A User's Guide to Spectral Sequences, Second Edition (Cambridge Studies in Advanced Mathematics 58)

0 we have bilinear mappings —

(p r : H(n,n +

H(q,q+

—

> H(n + q,n+q+r).

Suppose the following hold: (1) if n > q > ,n diagram commutes

+ r > n' +T.' and q+r > q' + r' then , the following on

H (n, n + 0 H(q,q +

j. H (n' ,

r') O H(q' ,

(2) For all n,

q > 0,r > 1,

r')

H(n+q,n+q+r) H(ri /

q/ ,

7,/ ).

S((p r (a 0 b)) = (pi(na o 6b) + (- 1)deg a (6a 0 no, where Ty and u are the appropriate maps in the Cartan-Eilenberg system.

Show that these data give rise to a spectral sequence of algebras. 2.4. Suppose that {D, E,i, j,k} is an exact couple of bigraded modules where bideg i = (1, 1), bideg j = (0,0) and bideg k = ( 1, 0). State and prove the analogue of Theorem 2.8 for such an exact couple leading to a spectral sequence of —

homological type.

-

60

2. What is a spectral sequence?

2.5. Suppose that the conditions for Theorem 2.6 hold and we define

Zf. = im(H(FP Al FP+r —> H(FP AIFP+1 .4)) = im(H(Fr'+' AIFPA) —> H(Fr Al Fr+1 A)) El; — Z ,17.13173:: Show that this determines a spectral sequence converging to H(A, d). (Hint: Look at the proof of Proposition 2.9.) 2.6. Suppose we are given a tower of submodules B2 C B3 C•••CZN C•••C Z3 C Z2 C E2

along with isomorphisms Z 1 /Z Bp /Bp+i , show that these data determine a spectral sequence whose tower of submodules is the one given. What kind of collapse results can you prove from this representation of a spectral sequence if you add the detail of keeping track of bidegrees? In the case of an exact couple, we defined Zf. = k' (im : DP-Er > DP") and Bf. = j(ker : DP —> DP — r+ 1 ). Show that Zr C Z and Bf, C .13,23+1 and that the differential dr : Zr_ 1 1137 BrI.137 c Zr_ 1 113, is induced by j o k and has the appropriate kernel. —

- 1

- 1

- 1

8

Determine conditions on an exact couple that guarantee that the associated spectral sequence is a spectral sequence of algebras. (Hint: Use a Cartan-Eilenberg system or consult the paper of IMassey541 if you get stuck).

2.7.

2.8. Prove the Universal Coefficient theorem: Suppose that A is an abelian group and (G, a) is a complex of free abelian groups with differential of degree —1. Then for each n > 0 there is a short exact sequence with

a

O—*

(C .9)

A

—

> H„(C.

A,

a ® 1)

Tor(Hri _ i (C., a), A) —> 0.

Here Tor(G, A) is the kernel of the homomorphism 1 0 6: G RA G 0 FA o where 0 —> RA -> FA -> A —> 0 is a short exact sequence with FA and RA free abelian groups. 2.9. Prove the homological analogue of Theorem 2.15 holds for a double complex of the form { M,, d' , d" } with d' of bidegree (-1, 0) and d" of bidegree (0, —1), satisfying d' o d' = d" o d" = d' o d" + d" o d' =0. 2.10. If p and q are relatively prime, show that Z/plE is a projectivelZIpq--module. 2.11. Show that the definition of Tor*R (K, L) for K and L in DGModR does not depend on the choice of resolutions.

2.12. Consider the category of differential graded modules over a ring R with differentials of degree —1 and morphisms of degree 0. Develop the notions of projective and flat modules, projective resolutions, and proper projective resolutions for this category. Prove that there is a Kiinneth spectral sequence for this category.

2.13. Deduce the Kiinneth theorem from the Kiinneth spectral sequence.

3 Convergence of spectral sequences "The machinery of spectral sequences, stemming from the algebraic work of Lyndon and Koszul, seemed complicated and obscure to many topologists. Nevertheless, it was successful ... G. W Whitehead

In Chapter 2, we find recipes for the construction of spectral sequences. To develop these ideas further we need to clarify the relationship between a spectral sequence and its target; this is the goal of Chapter 3. To achieve this goal, it is necessary to introduce more refined ideas of convergence. These ideas require a discussion of limits and colimits of modules and the definition of a morphism between spectral sequences with which one can express the relevant theorems of comparison. In the case of a filtered differential graded module, conditions on the filtration guarantee that the associated spectral sequence converges uniquely to its target. The case of an exact couple is more subtle and we develop it after a discussion of some associated limits. We express convergence results as comparison theorems that answer the questions: If two spectral sequences are isomorphic via a morphism of spectral sequences, then how do the targets of the spectral sequences compare? Need they be isomorphic? We end the chapter with some constructions and Zeem an 's comparison theorem that reveals how special circumstances lead to powerful conclusions. 3.1 On convergence

Theorem 2.6 tells us that a filtered differential graded module, (A, d, F), determines a spectral sequence and, if the filtration is bounded, then the spectral sequence determines H(A, d) (up to extension problems). We want to remove the restrictive hypothesis of a bounded filtration and still retain convergence to a uniquely determined target. We begin with the case of a filtered differential module over a ring R. Let (A, d, F) denote a decreasing stable filtration on (A, d), • • • c

FP+1 A c FP A c FP-1 A c • • • c A.

3. Convergence of spectral sequences

62

Notice that our first hurdle in understanding H(A, d) is the fact that an inclusion FsA c PA need not induce an inclusion in homology, H( c): H(Fs A) —> H (Ft A). We have defined (suppressing one of the bidegrees)

Zr, = FA n d' (FA)

and

Br, = FA n d(FP'

to get the tower of submodules 13r,c13c---cBgc-•-cZgc---c4c4,. The E00 -term of the associated spectral sequence is given by this tower as

Ego =

f/ur

.

For an arbitrary filtered differential module, we ask, how does this E -term relate to the desired target, H(A,d)? Does it relate to some other filtered graded R-module as well? To obtain the induced filtration on H (A, d) in Theorem 2.6, we defined Zgo = FA n ker d and Bgo = FP A n im d and showed that, for a bounded

filtration, zgo Bio'0 FPH(A,d)IFP+1H(A,d). Observe, however, that these modules zgo and Bgo need not come from the tower. We extend the tower to include them:

C 13fC • • • C

Ur Bg C

BC Zgo C

RZ,P,

c•••czfc Z.

The equality, Bgo = Ur Bg or FA n im d = Ur(FP..4 n d(FP—A)) can fail with d(x) if Us FsA A, for example, when there is an x in A _ U 8 in some FP A. To avoid this pathology, we can require that the filtration be exhaustive, that is, A = Us Fs A. In practice, this condition is satisfied by the

reasonable examples. Next we consider the submodule of infinite cycles. Definition 3.1. A filtration _V of a differential graded module, (A, d), is said to be weakly convergent if, for all p, Zgo = nr Zir?, that is, if FP A n ker d = nr(FpA n (FP+r A)).

Some simple conditions on the filtration imply that it is weakly convergent; for example, 1) the filtration is bounded above, that is, for each ri, there is a value s(n) with Fs(v ) A = {0} or, more generally, 2) np FA = {0 } . The proof of Theorem 2.6 needs only slight modifications to prove the following result.

63

3.1. On convergence

Theorem 3.2. Let (A, d, F) be a filtered differential graded module such that the filtration is exhaustive and weakly convergent. Then the associated spectral sequence with Ef' q HP+q(FP Al FP+1 A) converges to H(A, d), that is, FPHP-Fq (A, d) FP+1 HP-Fq (A, d). The definition of weak convergence seems dependent on explicit knowledge of (A, d, F). It would be helpful to have some equivalent formulations of weak convergence in terms of cruder invariants such as the homology of subquotients of A or the homology of stages in the filtration.

Proposition 3.3. The following conditions are equivalent on a filtration F of a differential graded module, (A, d): (1) F is weakly convergent. im(H(FA,Fp -H- A)_> H (FP+1 A)) = {0 } . n

(2)

r>1

(3) For all p, the mappings induced by the filtration, RP+1 —> RP, are im(H(FP-FrA) -> H (FP A)). rnonomorphisms, where RP =

nr

PROOF ([Cartan-Eilenberg561): Consider the commutative diagram of Rmodules where the bottom row is exact at B:

D

A

B

C

Since im k/ im f imk I ker g and im k/ ker g im(g o k), g induces an isomorphism: im kl im f From a filtered differential module we obtain such a diagram H(FPAl FP±rA)

H(FPA)

> H(FPAl FP+1 A)

H(FP+1 A)

induced by the long exact sequences that arise when homology is applied to the short exact sequences 0 FP+ 1 A —> FA —> FPA/FP+1 A —> 0 and 0 —> FP -FrA FA FPA/FP-Er A —> 0 and to the inclusion FP±r A c FP+ 1 A. To show that (1) is equivalent to (2), we begin with the observation that im{H(FPA/FP -FrA) —> H(FP+1 A)} is isomorphic to the quotient of im{H(FP AI Fr+rA) H (FP Al FP+1 Ail by the subgroup im{H(FP A) —> H(FP Al FP+1 A )} , which follows from the exactness of the

64

3. Convergence of spectral sequences

bottom row. The cycles in FPA/FP±rA are represented by those elements of FPA with boundary in FP-Er A. It follows that Zir? = FPA n d-1 (FP-Prit) is isomorphic to the module of cycles in FPA/Fr+rA and similarly, Zgo is the module of cycles in FA. From the canonical projections we obtain the composites Zg H (FP Al FP-Er A) —> H (FP Al FP+1 A) and Zgo H (FP A) whose kernels are expressible as modules of lower filtration and lower degree in the tower. If (2) fails, we can identify some nonzero element in Ir > 1

im(H (FP Al FP±r A)

H(FP+1 2-1))

and in turn some representative of it lies in nr ZI,? but not in Zgo and (1) fails. If (1) fails, then an element in nr zi?? not in Z 1 gives a class in H (FP Al FP+1 A) not in the image of H(FPA) and so (2) fails. To establish the equivalence of (2) and (3) consider the commutative diagram induced by the filtration: H (FP Al FP+r A)

H (FP+r A)

H (FP A)

I

H (FP+1 A)

If z C RP+1 = n, im(H (FP-Er A) —> H (FP+1 A)) then, by the exactness of the top row of the diagram, x E ker(RP+ 1 —> RP) if and only if x is in the intersection of the modules im{H(FPA/FP -FrA) H(FP+ 1 A)} for all r. Therefore, for all p, I r ?1

im{H (FP Al FP±r A) H (FP+1 24)} = ker (RP+1 —> RP).

and (2) is equivalent to (3). A weakly convergent filtration guarantees that the spectral sequence from a filtered differential graded module, (A, d, F), converges to H (A, d), in the sense that the E is related directly to a filtration of H (A, d). How do we know that the spectral sequence converges to H (A, d) and not to something else, say H (B , d)? To illustrate, let K be a free graded R-module and (A, d, F) a filtered differential graded module. Consider (A ED K, d (I) 0, F') with the filtration F' given by F'P(A e K) = FA e K. It is easily seen (check the associated tower of submodules) that the spectral sequence arising from (A, d, F) is the same as the spectral sequence coming from (A e K, d e 0,F'). Furthermore, if F is weakly convergent, so is F'. Thus the same spectral sequence converges to both H (A, d) and H (A, d) K A condition on a filtration that prevents this trivial but malevolent example is to require that the filtration be Hausdorff, that is, it is weakly convergent and np FP H(A, d) = {0 } .

65

3.1. On convergence

The terminology is meant to recall the function space situation where convergence of a sequence need not determine the limit. The Hausdorff condition will be developed in the later sections. Morphisms of spectral sequences

Given two spectral sequences, {(E*, dr)} and {(E*, d r)}, we define a morphism of spectral sequences to be a sequence of homomorphisms of bigraded modules, fr : (Er" , dr) (E* , dr ), for all r, of bidegree (0, 0), such that fr commutes with the differentials, that is, fr o dr = dr 0fr , and each fr+i is induced by fr on homology, that is, fr+i is the composite _

H(f ,

) > B- (E,*'* , dr )

dr)

fr+i:

The class of spectral sequences, with morphisms so defined, constitutes a category, SpecSeq. It is useful at times to consider certain constructions as functors from a topological or algebraic category to SpecSeq. Suppose we have a morphism, Ur {(Er, Jr)}, of spec(Er dr)} tral sequences. Recall that each spectral sequence may be presented as a tower of submodules of its E2 -term. By restricting f2: E2 —> E2, we get the diagram: } : {

B2 C B3 C

C

B2 C B3 C • • • C

The condition fr. induced by f2:

n

T n

Br c

n

c .•. c

Z3 C Z2 C E2 t2

Br c

n

0 dr = dr 0 fr allows

fr+1: Er +1

Zr

,

Zr /B r

Zr C • • C Z3 C Z2 C E2

us to identify fr±i with the mapping

Zr /Br

Er

.

E. Furthermore, such a morphism induces a mapping f: E The second condition, that fr± i is induced by fr on homology, can be expressed in the diagram 0

br

Cr

Er +1

0

br

Cr

Er+1

0

where Cr = ker dr : Er —> Er and br = im dr . With these observations and the Five-lemma it is easy to prove the following result (exercise).

66

3. Convergence of spectral sequences

Theorem 3.4. If {f r } : {(E,„", dr)} —> {(4 ,* ,dr)} is a morphism of spectral sequences and, for some n, f„: E, —> En is an isomorphism of bigraded modules, then for all r, n < r Er is an isomorphism.

Thus an isomorphism at some stage of the spectral sequences gives an isomorphism of E„-terms. Morphisms of spectral sequences arise in the case of filtered differential graded modules when there is a mapping 0 : (A, d, F) —> (A, d, P') with : A —> A a morphism of graded modules, such that 00d= do 0, and 0 respects the filtration, that is, 0(FPA) c FDA. Such a mapping is a morphism of filtered differential graded modules and this notion leads to an appropriate category. It is immediate that 0 induces a homomorphism of the associated towers of submodules and so a morphism of spectral sequences. Since the spectral sequences are taken to be approximations to H(A,d) and H (A, d), it is natural to try to compare these modules through the associated spectral sequences as in the following result of [Moore531. Theorem 3.5. A morphism of filtered di ff erential graded modules 0: (A, d, F)

(A, d, F),

determines a rnorphism of the associated spectral sequences. If for some n, 0,: E0 —> En is an isomorphism of bigraded modules, then 0r : Er —> Er is an isomorphism for all r, n < r < oc. If the filtrations are bounded, then 0 H (A, d). induces an isomorphism H(0): H(A, d)

PROOF: It suffices, by Theorem 3.4, to prove the last part of the theorem. Since the filtration is bounded, there are functions s = s(n) and t = t(n) so that we can write {0}

Fs

c Fs-1 fin c

c

Ft+1 n 7--rn c Ft Hn = Hn

where Hn denotes H0 (A, d) and E FPHP+q/FP+I- HP+q. Similar data hold for In (A, d) which we denote as H 0 . Since 0„ is an isomorphism, by the boundedness of the filtration, we have, for the same s = s(n), Fs'Hn = F .-1 H n F s fi n Eoo s-1,n-s+1 Loso-1,n-s+1 FIf ' . We can now apply induction downward to Hn and Hn : Consider the commutative diagram with rows exact 0

FP Hn

FP 1 Hn

11 (0)

0

> FP Hn

111 (0) >

FP-1 Hn

l 'n P+1

0

00. EP-1'n-P+1

) 0.

3.2. Limits and colimits

67

We apply induction on i in the expression p = s— 1 — i and so we assume that H(0) induces an isomorphism FP_Hri FPHn. Since 0„ is an isomorphism, by the Five-lemma, H(0): FP-1 Hn P -1 B" is an isomorphism. Thus, we arrive at Hn = Ft Hrt — Ft B-7-1 _ Hn (Compare this proof with Example 1.K.) 3.2 Limits and colimits

In Theorem 3.5, we relied on the boundedness of the filtration to extract the comparison of the homology modules. In order to treat more general cases, we seek less restrictive conditions on the filtration. We first introduce some important constructions. Definition 3.6. Given a sequence of morphisms in a category C, Ds+1

Ds gs Ds-1

an limit (or inverse limit) ofthis sequence is an object in C, D = lim {Ds , g 8 }, <_s together with morphisms h s : D" —> D' such that g, o hs = h8 _ 1 for all s, and, for any object E in C, together with morphisms j s : E Ds such that ° is = is—i, there is a unique morphism k: E —> D' so that, for all s, j, = h, o k. Dually, a colimit (or direct limit) of the sequence is an object in C, = lim{Ds , gs }, together with morphisms p s : D' —> D' such that ps _ l o gs = ps for all s, and, for any object E' in C, together with morphisms q s : D' —> E' such that q8 _ 1 o gs = qs , there is a unique morphism k' : —> E' so that, for all s, (is = k' o Rs .

Defined by a universal property, limits and colimits are unique up to isomorphism. When the sequence {D 8 , gs } consists of subobjects and inclusions, we have lim{Ds, gs } = fl D8 and lim{Ds, gs } = U Ds. Thus limits and <_8 —8 colimits generalize intersections and unions. We call a category (sequentially) complete if any sequence of morphisms has a limit and (sequentially) cocomplete if any sequence of morphisms has a colimit. In particular, the category of graded modules over a ring R is seen to be complete and cocomplete by letting lim{Ds,gs} =(... ,x8 ,x 3 _1,...) G <_s

fl

DS

for all s, g8 (x 8 ) = xs _i}

with hs : lim{Ds,gs } —> Ds the 8th projection. The colimit is defined by —s

lim{D', gs } = Us D8

3. Convergence of spectral sequences

68

where x

G D'

is related to y

E Dt

if there are integers n and m with

g8 —ri-k1oy8_n+2 0. • •ogs (w) = gt_ m_k1o,yt_m+20. • .0gt(y) E D s

= Dt

.

The morphisms ps : Ds D — Q° are given by taking the inclusion into the union followed by the quotient. Given two sequences of morphisms in C, say {Ds , 98 } and {Ds, *0 9 }, then a morphism of sequences in C is a sequence of morphisms in C, fs : Ds —> D 8 , for all s, satisfying f8 - 1 o gs = o fs . When C has notions of kernel and cokernel, then we can talk about a short exact sequence of sequences of morphisms in C. Among the important properties of limits is their behavior when applied to an exact sequence of sequences of morphisms. We restrict our attention to limits and colimits in the category of R-modules. Lentma 3.7. Suppose 0 —> {Ks , ks } —> {Ds , gs } —> {Qs ,g,} —> 0 is a short exact sequence of sequences of R-modules and module homomorphisms. Then on application of the colimit we obtain a short exact sequence of R-modules,

0

lim{K8 , k s }

lim{D', gs } —s

lim{Q' , qs }

O.

Furthermore, on application of the limit we obtain a short exact sequence, 0 —> lim{Ks, k s } —>

g8 }

—8

lim{Qs, qs }.

PROOF: We leave the proof of the exactness for the colimit to the reader. It is a simple exercise in definitions. lEilenberg-Moore621 compute the limit using the homomorphism

: fJ

D8

H

D s determined by pri o = P — 97,41 o pri+i for all n.

By construction lim{Ds, gs } = ker <-8

If we view (II : fl Ds —> HD' as a

chain complex, concentrated in degrees 0 and 1, then the exactness of the given sequence leads to the exactness of the short sequence of chain complexes 0

HK

0

K

11 Qs

S

lisps

H

Qs

->•

and passing to homology, we get the short exact sequence associated to the limit.

69

3.2. Limits and colimits

The failure of the exactness of the limit is measured by continuing the exact sequence associated to the homology of chain complexes further. The first derived functor of the limit was introduced by IMilnor621 and is defined by lim 1 Ds = cokernel of 43. <-8

This leads to the exact sequence 0 —> lim{K s, ks }

lim{Ds, gs } —> lim{Qs, q 8 } <- 8

—> 11 m 1 {K 8 , k 8 }

11m 1 {D 8 , gs } —> lim 1 {Q 8 , qs } —> O.

Given a fixed R-module A we can form the sequence of homomorphisms all of which are the identity i A on A. This is a constant sequence and it satisfies lim{A, iA} = A = lim{A, iA} and lim 1 {A, iA} = {0 } . More generally, a sequence of homomorphisms that is eventually all isomorphisms has limit the eventual abstract R-module. If a sequence consists entirely of epimorphisms, then the limit maps onto each Ds , * and the first derived functor vanishes. This has the amusing consequence that a sequence of epimorphisms with vanishing limit must be the sequence of trivial modules. Given the filtered graded module, (H(A, d), F), we obtain a sequence of graded modules together with the canonical homomorphisms H(A)

H(A)IFP+1 H(A)

H(A)I FP H(A)

H(A)IFP -1 H(A)

which induces a homomorphism u: H (A) lim H (A) I FP H (A), unique up <-p to isomorphism. Definition 3.8. A filtration on a differential graded module (A, d) is said to be strongly convergent (Cartan -Eilenberg561) or complete (IEilenbergMoore621) if it is weakly convergent and the induced mapping u: H(A) —> lim H(A)I FP H (A) is an isomorphism. <-p

Theorem 3.9. Suppose 0: (A, d, F) —> (A, d, F), is a morphism of filtered differential graded modules so that for some n, E r*,, * En* , * is an isomorphism. If the filtrations are exhaustive and weakly convergent, then 0 induces an isomorphism of associated bigraded modules 4 ,* (H(A, d), F) Eo"(H(A,d),F). If the filtrations are also complete, then 0 induces an isomorphism on homology, H(0): H (A, d)—,H (A, d).

70

3. Convergence of spectral sequences

PROOF: By weak convergence, we know that En" (H (A, d), F) E* and Since 0 induces an isomorphism of the E„-terms of the spectral sequences, the associated bigraded modules for (H(A, d), F) and (H (A, d), F) are isomorphic. We construct H(0) with the E„-terms of the spectral sequences and the assumption of strong convergence. First we show that the isomorphism —> induces for all p and all r > 0, E0"(H (A, d), F)

FP' H (A) I FP H (A)—> o_ FP' H (A) I FP H (A).

This follows by induction on r. Since FP-1 H (A)/ FP H (A) EV-* —>

the case of r = 1 is established. To complete the induction, apply the Fivelemma to the mappings between short exact sequences of the type 0 —> FP- r+1 H (A) I FP H (A) —> FP' H (A) I FP H (A) —> go-r+1 , * o

where the first isomorphism is given by induction and the last by the isomorphism ElZr+ 1, * Since F and F are decreasing and exhaustive, 0,0 induces an isomorphism, for all p, U FP- rH(A)IFPH(A)= H(A)IFPH(A)

->H(A)/H(A) = We can now pass this isomorphism to the limit to get the commutative diagram H (A, d) H(ç ) H (A, d)

U

> lim H (A) I FP H (A) <-p

lu > lim H (A) I H (A)

and by completeness, H(0) is also an isomorphism. This comparison theorem identifies the condition on a filtration of a differential graded module so that the associated spectral sequence determines the target module H (A, d) (uniquely up to extensions). It also offers the motivating example for strong convergence of more general spectral sequences. The key property that makes strong convergence work for a filtered differential module is an inverse limit condition. We next consider more general situations.

3.2. Limits and colimits

71

Completion

The property of strong convergence is determined by comparison: If we know that a spectral sequence converges to a filtered R-module, (M, F), then F'M/F'±1 M. Given Eo (M, F), when can we reconstruct M? A filtered R-module (M, F) may be endowed with a topology: Take as basic open sets the collection of cosets of the submodules Fs M . A filtered module (M, F) is complete (in the topological sense) if any Cauchy sequence in the topological space M converges, that is, if a sequence of elements { x„} satisfies the condition, for all s, there is an integer N(s) such that x n, — x, E Fs M for any n, m > N(s), then there is an element y E M such that, for all s, there is an integer Q(s) such that y — x n, E Fs M for any n > Q(s). We now prove the following properties relating the underlying algebra to this topology. Lemma 3.10. A filtered module (M, 1) is Hausdorff in the the topology induced by the filtration if and only if Fsm= {0}. The module is complete in this topology if and only if the mapping M — km M/FsM is an epimorphism. Il

ns

follows that a filtered module (M, F) is Hausdorff if lim Fs M = {0}, and is complete if lim' Fs M = {0}. Dually, a filtration is exhaustive if and only if <_s lim M/FsM = {0}. +s

PROOF: We leave the properties of M as a topological space to the reader. For the algebraic equivalents, consider the short exact sequence of sequences of modules where q.s : MIFsM MIFs'M is given by m + FsM 1-4 m F' M: 0 —> {Fs M , c} {M, id}

{M I M, q s } O.

When we apply the direct limit functor, then, by exactness, the filtration is exhaustive, lim +s M =M = M , if and only if +s MIF'M = {0}. When we apply the limit functor, we obtain the short exact sequence: 0 lim Fs M M lim M I M lim l M O.

Since the morphisms in the sequence of filtration submodules are all inclusions, the Hausdorff property is equivalent to the vanishing of first limit. With lim Fs M = {0}, vanishing of the first derived limit of the sequence {FsM} <_s is equivalent to completeness. Strong convergence of the spectral sequence associated to a filtered differential module (A, d, F) is a feature of the completeness of the filtration of H (A, d). As in analysis, the failure of completeness is overcome by completing.

72

3. Convergence of spectral sequences

Definition 3.11. To a module M with a decreasing filtration F we associate the completion of M with respect to F given by = lim M I P's M <-8

The completion is equipped with a mapping c: M la induced by the quotients M M I Fs M and called the completion homomorphism. The filtration F on M induces a filtration P on la given by P.k = 11m F.' MI Ft M. —t

Proposition 3.12. The completion of a filtered module (M, F) satisfies the

following properties:

(1)(la, E) is complete and Hausdorff. Fs M Ft M. (2) For all t > s, FS / (3) P is exhaustive if and only if F is exhaustive. PROOF: Consider the short exact sequence

0 —> FM/FM —>

I Ft la —> M Fs M

0.

Varying t we get a short exact sequence of sequences of homomorphisms. Apply the inverse limit functor to get O—# Fs1124- —> 1124- —> MIFsM —> lim 1 F la I Ft la —> lim 1 M Ft M —> . —t —t Since F is a decreasing filtration, the morphisms Fs If I Ft la —> Fs M Ft-1 M are all epimorphisms and so lim i FsMIPM = {0}. Thus the exact sequence —t reduces to

(3.13)

0 —>

1124- —> 11/ —> MIFsM

O.

Next consider the limit over s:

0 —> lim Ps1124- —> 112/

lim /14/FsM

—s

—s

lim 'Ps

—s

o.

Since 11m MiFs M = la, the morphism in the middle is an isomorphism and so the modules on the ends vanish. It follows from Lemma 3.10 that (112/, I') is Hausdorff and complete. Notice that the short exact sequence (3.13) implies that M/FsM la I Ps la and so, by the Five-lemma and the diagram, 0

Fs MI Ft M

O

Ps k/Ptk

>

I Ft la

I Fs M

>0

I Ft fl

.70-IPsk

> o,

3.2. Limits and colimits

73

we get fr8k Êtla FS M IFt M. Furthermore, if we apply the direct limit functor to (3.13) we get 0

Ut

—>1ÇI lim M1F8 M —>s

0.

By Lemma 3.10, (ici, I') is exhaustive if and only if lim M/Fs M = {0} if and only if (M, F) is exhaustive. 111 The dilemma that convergence of a spectral sequence presents is clarified by the completion. If we have only an isomorphism .Eg„* E* (M, F), is this enough to capture M up to extensions? The answer is NO in general, but it is the completion that emerges as the real target:

Proposition 3.14. Let 0: (M, F) (N, F) be a morphism offiltered modules. If the filtrations F and .F are exhaustive, then the following statements are equivalent. : (1(1, P) .'t) is an isomorphism of filtered modules, that is, i'9 11^4 —> N is an isomorphism for all s. (2) E0 (0) : E0 (M, F) —> Eo (N, .F) is an isomorphism. (1)

PROOF: If induces an isomorphism of filtered modules, then it induces the isomorphism of associated graded modules Eo (0) : Eo ( F) E0 (112 F) Eo(g-6) Eo(N,.fr)

Eo(N •F) • Suppose E0(0) is an isomorphism. This implies that, for all s, 0 induces

an isomorphism E*(9) ( 0): Fs MlFs+1 M

.Fs N .Fs+1 N.

By induction and the Five-lemma it follows that 0 induces an isomorphism — I' it I F8 +" M sNITs±"N for all n > 0. Taking inverse limits over n gives us that ci) induces isomorphisms, for all s, Fs .11^/I = lim Fs M Fs+n M

lim ,Fs N ,Fs+n N =

çt, —n

Finally, induces an isomorphism of direct limits,

74

3. Convergence of spectral sequences

Corollary 3.15. Suppose {f r}: {E" , d r } —>{E»",dr} is a morphism of spectral sequences and for some n, f n : E —> En is an isomorphism. Suppose Er converges to (M, F) and E r converges to (N, .F). Then f oo induces an isomorphism of filtered modules, f: , —> S

Corollary 3.15 generalizes the successful comparison theorems (Theorems 3.5 and 3.9) for filtered differential modules. In these cases the target modules were already complete and so isomorphic to their completions. We next turn to exact couples. A morphism between spectral sequence can arise via a morphism of exact couples, {V) 1, '02 }: {D, E, j,

—> {D,

k}

D and '02 : E E by which we mean a pair of homomoiphisms /pi : D compatible with the homomorphisms in the couple, that is, T, o I/4 = Jo o j, and k 0 )//2 = )/4 o k. Under these conditions, qPi and '02 = induce a morphism of the derived couples and hence a morphism of spectral sequences. In the topological applications moiphisms of exact couples can arise from functorial constructions. We saw in Chapter 2 (for example, Corollary 2.10), that the target of the spectral sequence associated to an exact couple can be more difficult to describe than the analogous problem for filtered differential graded modules. To an exact couple we associate two limit modules: D" , * —

<—s

il and D —

i}

Each module has a decreasing filtration given by = ker(D' , *

D s' * ) and FsD ",* = im(D" —

D - 00,8 )

In fact, these filtrations have very nice properties. Proposition 3.16. For an exact couple, the filtration F on the colimit is exhaustive. The filtration .F on the limit D" , * is Hausdorff and complete.

PROOF: Recall that the colimit is a quotient of the union of the Ds ,* and so any element in D — " ,* comes from some Ds , * before the quotient and thus lies in some Fs D — Qc , *. It follows that the filtration is exhaustive. To prove that the filtration .F on D 8 is Hausdorff, suppose that (•••,xs,x8_,,•••)En s "Fspc".

Ds , * is given by projection, to be in the kernel of h, one has x, = 0. Thus if an element of the limit is in all of the filtration submodules, then it must be the zero element and the filtration is Hausdorff. Since h8 : D", *

75

3.2. Limits and colimits

To prove completeness we consider the isomorphism im(D''* —> I ..Ts lim —> D8 "). Applying the limit functor gives lim <_s <_s By the universal property of the inverse limit, the limit on the right can be identified with D" , * and so the filtration is complete.

We generalize Proposition 3.3 on convergence for filtered differential modules to exact couples. Lemma 3.17. Let RP = exact sequence for each s: r

Fs D - 00,4 7 FS+1D-00,*

0

DP, *). Then there is a short

( ir:Dp+,,*

D 1 —8+

Egc;*

R8.

PROOF (IBoardman991): We apply Proposition 2.9 to rename the modules in C Es ' * . Since

the sequence. We begin with ker k

13*

=

=

we see that 13* c ker k c

H rj(ker ir-1: Ds,*

Z.

Ds—r+1,.)

Consider the short exact sequence

0 ker

Z, -*

ker k

Notice that Z ,. 8* = k'R8+1 by the definition of Rs+1 . Then k induces a mapping k: R8+' by k(a + = k(a). The image of k is given by Rs+ 1 n 1m k = Rs+1 n ker i and so the sequence Bg.*

R s+1

R8

is exact at Rs+ 1 . Furthermore, k may be taken as defined on Z* and so has kernel given by ker k. Thus the mapping to Rs*" coincides with the mapping in the short exact sequence for 13g.* c ker k c Z. We next rewrite B,S.* in terms of the colimit D – " , * to show

U

(ker

: Ds , *

= ker (ps : Ds , * —> D –, *).

The union of kernels certainly lies in the kernel of ps . If p8 (a) = 0, then in (a) = 0 E Ds+n, * for some n by the definition of the equivalence relation that determines the colimit. This implies a E ker in and we get equality. It

follows that 13.,9,* = (ker

76

3. Convergence of spectral sequences

Using the homomorphism j we obtain an isomorphism D"

I (ker j

ker ps)

I(

` Ds '

*1

j (key Ps)

_ ker

IB

Using the homomorphism ps : Ds , * —> D — "'* we get a short exact sequence 13 ->

/;1

1

(Fs+ 1 D —

*) —> Ds , * —> F

0,) ,*

s

I Fs+1 D — c° , * —> O.

*) _ p; i0j8+1 (Ds+1,7 ) _ kerps +imi = kerps + However, p;1 (Fs+1 ker j. It follows that Fs /F8+ 1 is isomorphic to Ds , */(ker j + kerps ) and so to ker j1.13* and the lemma is proved.

It follows from the lemma that the modules Rs tell us if we can identify E,0 in terms of the fi ltered module given by the colimit. Generalizing Proposition 3.3 we see that the spectral sequence associated to an exact couple converges to the colimit D —, * whenever the morphisms : R' R 8 are monomorphisms for all s. Conditionally convergent spectral sequences

g

[Boardman99] introduced a class of spectral sequences together with an invariant that aids in recognizing strong convergence intrinsically. Definition 3.18. The spectral sequence associated to an exact couple {Ds*, E8 ' k } is said to be conditionally convergent to D — ', * =limps, * , ,

=

<—s

= {0} and lim 1 {D",i,} = {0}. <—s

We associate to the tower of submodules • • • c 47. , * c 4.' *1 c • • • the first derived module of the limit of this sequence of inclusions denoted by RE = lim l

.

The notation hides a nice property of 11 m1 : If (M, F) is a filtered module with N a submodule that satisfies N c Fs M for all s, then (F811/ N) <—s

—

i/N and lim 1 Fs M lim (F8 M/N) •

This follows from the short exact sequences 0 N F8 .71/1 Fs M I N Zp* = z, we can write RE* = O. Since Bg'6* C Z,S. , * for all r, and lim <—r lim 47:* /Bs,*. The strength of these notions is born out by the following result. For the motivating topological application of this result see the discussion of [Adams74, Theorem 8.2 1 .

3.2. Limits and colimits

77

Theorem 3.19. Suppose {D 8 ' 8 , E88 ,i, j,k} is an exact couple satisfying Es , * = {0} for all s < O. Suppose further that the associated spectral sequence converges conditionally to D' , *. Then the spectral sequence converges strongly to D — " , * if and only if RE* = {0 } for all s.

PROOF: We add a little more detail to the objects involved. Let Imr DS , * =

: Ds+r, *

Ds , * .

These submodules are organized in two directions: Imr+1 Ds , * c Imr Ds , * for all r, and, fixing r, the homomorphisms i map Imr Ds , * to Ina' D8-1, *. In fact, i(Imr D 9,8 ) =Imr+1 D8-1, *. This leads to two limits and

lim{Imr D", c}

lim{Imr Ds'*, i}.

Lemma 3.20. For all r and s we have lim Ina' Ds , * = n InirDs,*= 118 , -r

lim{Im" D 8, *, i} = lim{Ds , *, i}, and

km 1 {Imr Ds , * , i} = lim 1 {D 8, *, i}.

PROOF: The first assertion follows from the sequence of inclusions and the definition of RS. For the second assertion we work by induction. For r = 1 we have the short exact sequence 0 —> Tm' Ds:* D8

'

D"/Im 1 Ds,* —>

O.

These fit together into a short exact sequence of sequences where the homomorphism between the last modules are all the zero homomorphism. Since lim{./14„, Of = {O} = 01 for any sequence of zero homomorphisms, <_s <_s the exact sequence relating lim and lim 1 shows that the limit and first derived <-8 limit of {D 8,3 , i} and {Iml D 3 ' 3 , if are isomorphic. Since Im 1 (Imr Ds , * ,i) = Im r+1 D 8 ' 3 , we prove the second assertion by induction. —8

Next we consider the sequence • • • limit may be written

Rs+1 —> 118 —> • • • . The inverse

lim{Rs, i} = lim (lim{Imr D", c}, . <-8

<-8 <-r

(It would be nice to simply commute the indices, but ... ) We prove directly that lim{R 9 , i} = lim{D 8, *, = D°°' 3 . Consider the projections h8 : D", * —> Ds , * . Notice that the image lies in R s: An element of D`)° ,3 may be written as (... , x s+i , x,„ x s _ 1, ) with Imr Ds , * for all r. This x, = i(x 8±1) = i 2 (x 8± 2) = • • • . Thus x, lim{Rs, Furthermore, fact induces a mapping D 00,8 the inclusions —s

3. Convergence of spectral sequences

78

118 c Ds , give a mapping of sequences and hence a homomorphism lim Rs —>

lim D8,8 . By the universal property of limits we factor the identity L';90° , * lim Rs —> D°° , * and so the mapping lim Rs —> D°° , * is onto. Checking at 4-8

the level of elements, we see that it is also one-one having been induced by inclusions. Thus lim Rs lim Ds , * . From the definition of Ina' D 8 '*, we have that Zs,* = Imr-1 D 8 + 1, * and there is a short exact sequence: 8

> Z* I ker k

Imr-1 D +1,*

—

>I— mr D 8 ' 8 —> 0.

If we apply the functor urn, we get the exact sequence <—r

o

I ker k

Rs +1

Rs

—> RE" —> lirn Irrir

lim Imr D8 ' 8 —> O. <—r

We splice in the short exact sequence from Lemma 3.17 to get () FD – "' * /F8-kiD-00,* e5

RE* 00

L R8 Imr-1 D8+1 '* L lim Imr D" —> 0.

E 8

f8+ 1

Suppose REg,;* = {0 } for all s: From the exact sequence we find that j: R 8+1 -> 118 is an epimorphism for all s. By the definition of conditional convergence {0 } = D°° ,8 = lim R 8 and so we conclude that R 8 = {0 } for all s. Thus

Fs D °° '* /Fs+1D* (D – °° , *, F).

E.,98

and the spectral sequence converges to

We next show that a right-half-plane spectral sequence with RE* = {0 } for all s and conditionally convergent to D',8 is, in fact, strongly convergent, that is, (D –°° , *. F) is complete and Hausdorff. This is equivalent, in the . language of limits, to lim Fs D – °° , * = {0} = Ern' Fs A right-half-plane spectral sequence is given by the condition D 8,8 = D°, * for all s < 0, which implies that D',8 = Do, * and Fs = im j 8 : Ds , * —> D° ,* = Ims LA*. It follows that lim F8 D°°' 8 = lim Ims DO, * = R° = {0 } .

Therefore we have the Hausdorff condition. To study the derived limit consider the short exact sequence of sequences of homomorphisms: 0

ker ir

j

)inar D"

-0

lc 0

> ker ir-1

>

Ds+r-1,*

> irrir-1 D 8,*

>0

79

3.2. Limits and colimits

This leads to the exact sequence 0 —> lim {ker ir, il —> lim{Ds+r, *, il —> lim{Imr Ds, * ,C} —r

—r

L111 1 {ker ir ,

<—r

lim 1 {Ds+", <—r

lim 1 {Imr Ds , * , c} <—r

O.

l {Ds+r, * ,i} = {0 } and so the exact seConditional convergence implies lim <-r quence forces lim 1 Imr Ds ,* = {0 } . In particular, lim 1 Imr Du ,* = {0 } and <-r <-r completeness follows. Rioardman991 discusses more general situations and applications. The insight of conditional convergence is that the invariant REe' may vanish for reasons easily at hand; for example, when all of the modules E2" are finitely generated, or there are only finitely many nonzero differentials at any given (s, t). Thus the raw data E2" can hold the key to RE* = {0 } , an intrinsic condition with global consequences. Conditional convergence is a global condition that depends on the behavior of certain limits, and hence may be satisfied by virtue of certain structural features of the construction of a spectral sequence.

The Mittag-Leffler condition

Other global properties of the modules in an exact couple determine the behavior of limits and hence convergence of the associated spectral sequence. For example, if R = and, for every s, D8,8 is a finite abelian group, then the groups Im r Ds , * = im(ir : Ds+" —> D")

must become constant for r large enough. It follows that lim 1 {Imr Ds , * c 1 = <-r {0} because the sequence of inclusions stabilizes to equalities. Furthermore, (i1r_1 13 since Z:. , * = 8+1, *), we get a constant sequence after some finite = = • • • and it follows that RE* = lim ZS,* = index r, Zys. ,8 = <-r {0 } . Putting these data into the exact sequence of the proof of Theorem 3.19, we get a short exact sequence —> F813—°° ' I Fs+1 D-00,*

Rs+1 —> R s —>

O.

(Recall that Rs = lim Imr Ds = fl Imr Ds.) A generalization of these phenomena for R-modules is implied by the following condition (introduced by [Dieudonné-Grothendieck611). Definition 3.21.

A sequence of homomorphisms • • •

13 8-1, * —> • • • satisfies the Mittag-Leffler condition if for every s, there is an integer 'y(s) such that, whenever t > -y(s), we have

im(Ds±" —> a") = im(D5 +7(8) '* —> D8'8).

80

3. Convergence of spectral sequences

We can define the Mittag-Leffler condition more generally and allow 7(s) to depend on the codegree as well, that is, for every (8, t) there is an integer -y(s,t) such that, whenever n> -y(s, t), we have im (DS+71,t

DS,t)

im (DS-H(S,t),t

DS,t) .

For ease of discussion we suppress the codegree. The saine argument for the case of finite abelian groups proves the following result. Proposition 3.22. Given an exact couple {Ds*, Es , * ,i, j, k}, if the sequence of morphisms • • • —> Ds+ 1, * —> Ds , * —> • • • associated to the exact couple satisfies the Mittag-Leffler condition, then there is an exact sequence ' Fs-H

0 —> Fs13

—> R s+1

Rs —> O.

Corollary 3.23. The Mittag-Leffler condition for a sequence • • • —> Ds , * —> —> • • • implies lim i- Ds , * = {0}.

<= s

PROOF: Following Rioardman99] we introduce the bigraded R-module = f

im(Dt , *

D 8, *) = Imt-9 D8 ,

t

for t > 8, for t < s.

Whenever s > u and t > y, there is a homomoiphism /s , ' —> /u , ' determined by the commutative square D"

Ds'*

Du'*

Fixing t, we get a sequence of homomorphisms:

It follows that lim /s , ' = lim as,* and lirn 'is ,' = lim 1 Ds'*. To compute the inverse limit and derived fiinctor we consider the exact sequence: (3.24)

0 —> lim s —>

Ts 't -> 43 H Ts't

lim rs,t

81

3.2. Limits and colimits

Since the first term in this exact sequence is independent of t, it follows that lim lim /s , t = {0 } . We now apply two facts (Eilenberg -Moore621) left to —s the reader to prove: (1) lim fJ/s,t = fl lim /s't and 11m 1 11 11m 1 ] ,t s —t (2) If 0 — {AS} —> {B 8 } —> {Cs} —> {Ds} —> Ois an exact sequence of sequences of morphisms with lim i- As = {0 } , then there is an exact sequence <_s 0 lim As

lim Bs

lim Cs

lim Ds

—> lim 1 Bs —> lim 1 Cs —> lim 1 Ds —>

O.

Apply the functor linti to the four-term exact sequence (3.24) and we get an exact sequence from facts (1) and (2), n "

Fixing

um ps 18,tH

ns s

s,

1;

18 t ,

,H 9 —t lp,t

1m its,t 9 1—t

0.

the Mittag-Leffler condition gives the sequence of morphisms, imry(s)±n Ds,* _> Tm

8)+n-1

Ds,* _>

Im7( s) Ds , *

• • —> Imr Ds , * —> •

from which we conclude lim /s't = Ini1(8) Ds , * = This gives the exact sequence 0 -> 1 1m D8, * -> 1-1 Since

1n8

s

R 8 +1 —> Rs lim -DS,* = {0 } .

R'

Rs

is an epimorphism for all

—

Rs

and lim 1 /8, t = {0 } .

> lim l D

—

> O.

s, coker (11 = {0 }

and so

Corollary 3.25. Suppose {D 8 ", Es , * i, j, k} is an exact couple such that the sequence of morphisms • • • —> Ds , * —> D 8 +1, * —> • • • satisfies the Mittag- Leffler

condition, then the topology induced by the filtration on lim Ds , * = D —, * is complete.

PROOF: The filtration on D — ', * is given by Fs D — ', * = im(Ds, * —> Ps This leads to the short exact sequence 0 —> kerp s —> Ds , * —> imps —> O. Applying the inverse limit functor gives the exact sequence that ends with lim 1 D 9, * —> lim imps —> 0 and, since 11m 'D' = {0 } , we have that —9 —9 11—ms 1 imps = lim Fs D — ", * = {0 } and so the filtration is complete. —s

82

3. Convergence of spectral sequences

Corollary 3.25 tells us that the only thing standing in our way from identifying the target of strong convergence of a spectral sequence in the presence of the Mittag-Leffler condition is lim F' , that is, whether or not the filtration _s on the direct limit of {Ds , *, il is Hausdorff. Like the point-set interpretation of the Hausdorff condition, it determines if the spectral sequence converges to a unique R-module or not. Notice that lim Fs D — ", * is also the kernel of the completion homomorphism when lim 'FsD — " , * vanishes. <_ s Recent studies of the convergence of spectral sequences have concentrated on the spectral sequence associated to a tower of fibrations introduced by [Bousfield-Kan721. The associated towers of groups fit into the theory of pro-groups and pro-morphisms that emerged from the study of étale homotopy theory (Artin-Mazur691, [Qui11en691). The basic reference for this theory as applied to homotopy theory is the book of Mousfield-Kan721. General convergence results for the homology spectral sequence associated to such towers are due to Hiwyer751, [Bousfield871 and to [Shipley96]. 3.3 Zeeman's comparison theorem

The next theorem shows how limited information can be turned into global information about a spectral sequence. Though the hypotheses seem specific in their algebraic formulation, the geometric interpretation of these results in Chapter 5 carries great generality. This theorem has become a standard tool in the application of spectral sequences. Theorem 3.26 (Zeeman 's comparison theorem). Suppose {E* ,d r } and {E* , are first quadrant spectral sequences of cohomological type (that is, dr and dr of bidegree (r,1 — r)). Suppose {L.} is a morphism of spectral sequences and the following diagram commutes with rows exact:

0

—>.TorR, (Er ,o E20,q)

4° e E2° ,q f20f2

g

' ()

f2I

E2° ' q

Tor( f2, f2

TorRi (g+1'° , E2Qq)

0

Then any two of the following conditions imply the third: L f2 :

EV

—>

f2: E2c" —> IlLf oo :

Ela q

—

is an isomorphism for all p. is an isomorphism for all q. > Elc3;q is an isomorphism for all p, q.

PROOF: That I and II imply III follows from the functoriality of TorRi from which it follows that Tor ( , f2) is an isomorphism and so, by the Five-Lemma, is an isomorphism for all p,q. Then apply Theorem 3.4. t2: —>

83

3.3. Zeeman's comparison theorem

We establish that I and III imply II by sneaking up on II by induction. Let Ilk designate the statement: f2: e'q —> e' q is an

isomorphism when 0 < q < k.

Since E20 ' ° = E°, III establishes Ho; we assume I, III and ilk. We first prove some facts. Fact 1. fr : EP — f when q
is an epimorphism when q < k and an isomorphism

Observe that Fact 1 holds when r = 2 by the Five-lemma, I and Ilk. Suppose it is true for r = m. Consider the following diagram with the rows exact: (ker dm )P''

0

f

I

(ker dm )P'q

0

E

Ep +m , q—m+ 1

Efr;,g

Ep+m, q -m+1

If q < k, then q — m 1
Let fi G (ker d.„)P'q. This goes by inclusion to ft E E. Since fm is onto, there is an element u E E with fm (u) = ft. Since f, is an isomorphism on the right, we have dm (u) = fm-1 0 d, f ,(u) = fm-1 0 d,('t1) = 0,

so u is in ker dm and fm (u) = ft. Thus 1, is onto. If q < k — m+ 2, then both morphisms f, are isomorphisms. If u, V E (ker dm )v satisfy fm (u — y) = 0, then fm (u — 21) = 0 and so u= y in Since (ker dm )P , q injects into EV, u = y in (ker dm )v and so fm is an isomorphism. Consider the dual diagram: EP—rn,q+m-1

fm 1

d,

.0

land,

0.

)

)

fm

Ep—Trt,q+M-1

Once again, since f, commutes with the differentials, the homomorphism

aig Imd, —>

im d,

is induced by fm.

84

3. Convergence of spectral sequences

Let U E EVj im dm . Then there is an element r with Q(C) =U and U' E EV. When q < k, there is an element U' with fm (P) = U', since fm is an epimorphism. Let U = Q(U'). Then fm (U) = U and we see that fn., is onto. Suppose q < k — m + 1. Then q + m — 1 < k and the leftmost fm is an epimorphism, the next f, an isomorphism. Suppose U, V E EV/ im 42 satisfy in, (U — V) = 0. Suppose U = Q(U'),V = Q(V'). Then, Q o fm (EP — V') = fm (U — V) = 0 and so there is an element W G gri—m 'q+m—1 with dm (W) = fm (U/ — V'). Since W = fm (IV) for some W E Efjrn , g+7n —1 , we have dm (W) = — V' and U

—

V = Q(U' — V') = Q O dm(W) = 0.

Thus f, is an isomorphism. Next consider the diagram 0

> (im dm )P'q

0

(im dm )P'q

>

dm

>0

lim dm

0.

When q < k, fm is an epimorphism, and we can conclude fm l is an epimorphism. When q < k — m + 1, both fm and fm are isomorphisms and so the argument for fm works to show that fm is an isomorphism. Finally, let us consider the diagram: (im dm )P'q

0

> (ker dm )P'q

fml 0

1m

>

frn+11

(ker dm )P'q

(im dm )P'q

> EPm+1

EI;a1 —>.

o.

If q < k, then 1, is an epimorphism and hence so is fm+1 . If q < k — rri + 1 then fm and fm are both isomorphisms and so then is fm+i. Since k —m +1 = k — (m +1) + 2, we have established Fact 1.

Fact 2. When r is large enough, E,..° ,7E1 = EY+1 and so fr%ki+1 = g8k -kl an isomorphism.

This follows because we have a first quadrant spectral sequence. From Fact 2 we argue that 49 , k+ 1 is an isomorphism and, descending to r = 2, establish _14+1. By its placement in the first quadrant, Er(W 1 = (ker dr) 0,k+1. Consider the diagram: (ker dr )o,k+1

Ero,k+i

Err,k-r+2

f,

0

(ker dr )o,k+

E1'

Err,k-r+2 im dr

f,

d,

k r' k-r+2 r

Er' k—r±'2 411 dr

O.

3.3.

85

Zeeman's comparison theorem

an ,somo isomorphism. By Fact 1, Jr We have „f r — jr+ i — J o° fr is n isomorphism and so is the rightmost fr. If 'a E Er0 ' k+1 , then ft, = fr (u) = fr (u) when dr (d) = O. Suppose dr (ft) = E Err, k—r+2, O. Then f.) = fr (y) since fr is an isomorphism. But Q (v) = f» o (2(v) = 0 and so y = dr (v) for some u E E,9, ,k+1 . Thus fr : —> L,9, , k+' is onto. When u, y E 43,k +1 satisfy fr (u — y) = 0, then dr (u — y) = f,7 1 0 dr 0 fr (u — y) = 0

so u — (ker dr )° , k+ 1 . Now fr (u — = fr (u - v) = 0 implies u = y and is an isomorphism and IIk±i holds. so : E,9 , k+1 The proof that II and III imply I is similar. We next give an application of Zeeman's theorem that generalizes the examples in §1.4. We remark once again, though the next theorem seems to be a quite specialized piece of algebra, the applications of it are spectacular. Let R = k, a field, and suppose that the following hypotheses hold: Hypotheses. (1)

dr} is a first quadrant spectralsequence of algebras with E2* ' * that is, V* E 2* '° and W* E 2° ' , both as algebras. Eco is the bigraded k-vector space k, that is, El° k and Eg2 = {0} for (p, q) (0, 0). W* = an exterior algebra on generators xi , x9, . ,x, where each xi is homogeneous of odd degree 2ri — 1. O. (Anticipating For each i, di (xi) = Ofor 2 < j < 2ri and d2,(xi) terminology, we say that x i is transgressive; see §6.1.) char k 2 (a technical detail regarding exterior algebras). V * Ok W * ,

(2)

(3) (4) (5)

Theorem 3.27 (More1531). Under the hypotheses (1)—(5), V* is isomorphic to k[yi, Y2* • • • , yrn] where yi = d2,(xi ).

PROOF (Veeman58]): Since we are working over a field, we have a definition of tensor products of spectral sequences. It is easy to establish the following: a) if E and E have E„ -terms isomorphic to k, so does E E;

b) if E* , *

V* OW*

and E* , *

V* ®W*

then (E ® E )* , *

(V V*)

(W W*):

c) an associative product for E induces a morphism of spectral sequences, for

each n,E0E0-- .0E-E. n copies

86

3. Convergence of spectral sequences

Let s be an odd integer. We introduce E(s), the elementary spectral sequence over k of degree s, given by E(s) 2 = A (u, ) k[v, +1] with deg u, = s, deg vs+ 1 = s + di = 0 for j s + 1, and ds +i(us ) = vs +1. Because the action of the differential d8 _1 is determined by the Leibniz rule, E(s),,, k, with the pattern of d,±1 given in the diagram.

A:

By hypothesis 4 we can construct a unique mapping, for each i, 1 <j < m, E(2ri -1) 2 E2, which extends to a morphism of spectral sequences, by

fi (u2ri _ 1 ) = xi and fi (Y2ri) = d2r i (xi) • Since E2" *=' V * Ok W* , everything else about xi is determined by d2, (xi) = O. Collecting these morphisms for i = 1, 2, ... , m, we obtain the composite F: E(2r i — 1) 0 E(2r2 — 1) 0 • • • 0 E(2r, 1)

f1ef2

•efr.

multiplication

E0E0•••OE

> E,

a morphism of spectral sequences. By definition E(2r 1 —1)0. •

•®E(2r,-1)'=' A(u2r1-i,••• ,u2,-__1)ek[v2ri,

• • • ,v2,-,]

and F, at the E2-term, maps A (u2r i —1, • • • , u2rm _1) isomorphically to W* = A(x , , x m ); thus we conclude that the tensor product of elementary spectral sequences satisfies part I of Theorem 3.26. Furthermore, since each spectral sequence has Em -term, k, F induces an isomorphism of the Eco -terms and part Ill of Zeem an 's comparison theorem is satisfied. Thus, F induces an isomorphism k[v2r i , • • • V* and the theorem is proved. For a discussion of the case of char k = 2 see the paper of [Zeeman581. The technique in this proof of comparing an algebraically constructed spectral sequence with an already existing one is standard and has wide applications. Important applications of Theorem 3.27 appear in the work of [Bore1531 on the cohomology of Lie groups and their classifying spaces (see Chapters 5 and 6).

Exercises

87

Exercises 3.1. Suppose (A, F) is a filtered module over R so that F is bounded and each

E

(A, F) in the associated graded module is projective over R. Show that A is determined, up to isomorphism, by these conditions.

3.2. Show that a morphism of filtered differential graded modules leads to a morphism of the associated spectral sequences. Show that a morphism of exact couples determines a morphism of the associated spectral sequences.

3.3. Suppose (M, F) is a filtered module over R. Topologize M by taking all cosets of all submodules FPM as basic open sets. Show that the filtration is Hausdorff if and only if M is Hausdorff in this topology. Show that the filtration is strongly convergent if and only if every Cauchy sequence in M with this topology converges in M. This completes the proof of Lemma 3.10.

3.4. Suppose that [Er* , * , dr } is a first-quadrant spectral sequence, beginning with the El-page. Suppose further that the differential graded module, (tot al (E1* ' * ), dj) is filtered and dj respects this filtration. Show that there is a spectral sequence, derived from the filtration of tot al(E * ), converging to E2*,* of the original spectral sequence. Show further that this new spectral sequence inherits a third grading from the bigrading of Er. This spectral sequence is sometimes called the algebraic May spectral sequence ([May64]).

3.5. Prove Theorems 3.2 and 3.4. 3.6. Suppose }Er* , * , dr } is a first quadrant spectral sequence over a field, k, and Er converges to (H* , F). Let V be any graded vector space over k and filter H* d)V by PP (II' e v) = FPH. e V. Show that Er converges to (H* e v,P). Suppose (H* , F) is such that F is strongly convergent. Show that (H* El) V, is not strongly convergent. This shows that strongly convergent implies convergent but not vice versa.

3.7. Show that the definition of the inverse limit and direct limit for sequences of modules over a ring R satisfy the defining universal properties. Show that the direct limit functor is exact (Lemma 3.7). Finally, show that a sequence of epimorphisms of R-modules with vanishing inverse limit is a sequence of trivial modules.

3.8. Let f and g be morphisms of filtered differential graded modules,

f , g : (A, d, F) —> (A, d, F) and suppose f is chain homotopic to g via s : A —> A of degree —1, that is, ds sd = f — g. We say that S is a homotopy of order k if s(FPA) C Prove that, if a chain homotopy of order k exists between f and g, then for r > k, the induced maps on the spectral sequences satisfy f r = gr and fc.o = goo.

3.

88

Convergence of spectral sequences

3.9. A Rees system ([Eckmann-Hilton66]) or spectral datum (Wilenberg-Moorep is a diagram of modules over a ring

R )

D

D

in which {D,E,i, j, k} and {D, E, T, j, k } are exact couples, all triangles commute and the sequence

D D F 1>rD is exact. Show that the derived couples of a Rees system give another Rees system in which F remains the same. Show that if two exact couples are part of a Rees system, then their associated spectral sequences are identical. The motivating example is given by (A, d, F), a filtered differential graded module: Derive the Rees system:

H(FP+1 A)

H (FP A) N

H(A)

H( FP Al FP+1 A)

H (Al FP+1 A)

H (A)

) H(AIFPA)

3.10. Suppose (M, F) is a filtered module with N a submodule that satisfies N C

Fs M for all s. Show that lim (Fs M/ N) (lim Fs M) /N <-8

)

and

lim 1 Fs M <-8

lim 1 (Fs H/N)

3.11. Prove the assertions (1) and (2) of [Eilenberg-Moore621 used in the proof of Corollary 3.23. 3.12. Prove the assertions a), b), and c) about tensor products of spectral sequences made in the proof of Theorem 3.27. 3.13. Let {M" , d' ,d 1 } denote a double complex where there is no restriction on the bidegrees. Apply the notions of convergence of Chapter 3 to find conditions under which the associated (planar) spectral sequence converges (weakly, strongly, conditionally) to H (tot al (M * , D).

Part II Topology

90

4 Topological Background "Of course, one has to face the question, what is the good category of spaces in which to do homotopy theory?" J. E Adams

The classification of topological spaces up to homotopy equivalence is the central problem of algebraic topology. The method of attack on this problem is the application of various functors from the category of topological spaces (or a suitable subcategory) to certain algebraic categories. These functors do not distinguish between homotopy equivalent spaces, and the algebraic data they provide may be enough to distinguish nonequivalent spaces. In order to make the problem of classification more reasonable, it is necessary to identify a tractable category of spaces. A candidate for such a category should be large enough to contain all 'important' spaces (such as finite-dimensional manifolds) as well as contain the results of various constructions applied to these spaces (for example, it should be closed under suspension, loops, etc.). Furthermore, the classical homotopy functors, singular homology and cohomology and the homotopy groups, should be effective in distinguishing spaces in this category. In this chapter, we present two categories of spaces that satisfy the desiderata of homotopy theory. In §4.1, CW-complexes are defined. The homotopy groups of CW-complexes are effective enough for a classification scheme, and the combinatorial structure allows the computation of their singular homology groups. In §4.2, the equivalent category of simplicial sets and mappings is presented. This category features a more rigid combinatorial structure in which the classical homotopy invariants can be defined and studied. Furthermore, the general notion of a simplicial object over an arbitrary category leads to many useful constructions in homotopy theory. Both a CW-complex and a simplicial set give rise to filtered spaces and so the algebraic technology developed in Chapters 1, 2, and 3 may be applied to the computation of the homotopy invariants of such spaces. Another way in which the homotopy invariants of a space may be restricted enough to allow computation (or at least the construction of a spectral sequence) is to place the space in a system of spaces and maps with good exactness properties with respect to homotopy functors. Exactness properties can be translated

92

4. Topological Background

into the problems of extension and lifting. Mappings with good extension properties are called cofibrations and they lead to sequences of mappings that give exact sequences when the functor [—, Z], homotopy classes of maps to some space Z, is applied. The dual notion of good lifting properties is enjoyed by fibrations (§4.3). Exact sequences result when the functor [Z, —] is applied to a libration and, in particular, if Z = Sn, these exact sequences consist of homotopy groups. By introducing a space 'dual' to an n-cell (with which one builds a CW-complex), one can construct a tower of librations that approximates a space one homotopy group at a time. The cell decomposition of a space (reflected in its homology groups) and its Postnikov tower (which carries the homotopy groups) are sources of homotopy invariants and so figure in the constructions found later in the book. We finish this chapter (§4.4) with a discussion of some of the further structure that is available on singular homology and cohomology. With coefficients in a field, the dual of the cup product structure for cohomology is a coalgebra structure in homology. In the presence of a multiplication on a space (an H-space structure) such a coalgebra structure arises on cohomology and is intertwined with the cup product structure. For an H-space of finite type, the homology and cohomology are dual Hopf algebras, and the structure theory of Hopf algebras may be applied. The algebraic aspects of Hopf algebras play an important role in later chapters. The cohomology of a space with mod p coefficients enjoys another significant piece of structure; H* (X; i s an algebra over the mod p Steenrod algebra, ,Ap , of stable cohomology operations. The applications of this structure have led to some of the deepest results in homotopy theory. We record some of the basic properties of the Steenrod algebra in §4.4. This chapter plays two roles in our exposition. First, it supports the constructions found in later chapters and so affords the reader a ready reference for topological facts that would take the reader too far afield to develop in the places where they are needed. Secondly, it provides the novice a thumbnail sketch of some of the fundamental constructions in homotopy theory. In both of these capacities, it is distracting to overburden the reader with details. Therefore, we omit most proofs and give copious references for the reader who needs details. The reader who is acquainted with these standard notions should skip to the next chapter and refer back when necessary. 4.1 CW-complexes

Let's begin by building a topological space. For building blocks, we take the n-cell, Pn = Rn < 11. Note that the boundary of the n-cell, 0e, is the (n — 1)-sphere, Sr'. We proceed a dimension at a time. In dimension 0, let x° be a finite set of 0-cells (points). We next take a collection of mappings : S° = ael —> X(°), j = 1, . . . , m, called the attaching maps, and form the adjunction spaces, X,(o) = (°) Uf„ el,

4.1. CW-cornplexes

93

,(co , , uf„ e 1 , and so on to X (1) -- Xn(°, 11 Uf, . Recall that a mapping g: S —> Y gives rise to the space Y U, en+ 1 determined by the v

quotient space

Y

en ki -

I {X

g(x) for x E S n = 00+ 1 1 ,

where LI denotes the disjoint union. In dimension n, assume X (31-1) has been built and take a collection of mappings, fn.' : 5" -1 = ae" X ("—l) , for j = 1, . , r. Form the adjunction spaces, 4 71-1) = X (n-1) Ufn„ en , 4n-1) — — 4n-1) Uf 2 e n, and so on, tip to X(31 ) = Ufr, en . If we continue to add higher dimensional cells, the space X that results in the direct limit (union) of this process is called a cell-complex. This notion was introduced by [Whitehead, JHC491 as a natural generalization of polyhedra. Spaces that can be built in this fashion are characterized by the following point-set description:

Definition 4.1. A CW-complex is a topological space X, together with a cell decomposition {Oa E A}, so that, for each n and a, ena C X, and (1) X is the disjoint union of the collection {int ea } . (Here int denotes the interior of the cell and int e0 = e° .) (2) ena is closed in X. (3) Let X(n) = H n em a ; the space X (n) is called the n-skeleton aeA, m<

of X. The successive skeleta filter X; that is, there is a sequence of

subspaces, X (°) c X (1) C

C X (n) C

...X.

(4) For each a E A, there is a mapping, fa : (en ,Sn -1 ) —> (Oa , 00,), such that f a l: int en —> jut ena is a homeomorphism; f, is called the characteristic mapping for the n-cell, ena. (5) cls(ena ) is contained in finitely many cells (we say X is closure finite). (6) K C X is closed in X if and only if for all a, K n cis(e) is closed in int(ena ). This is called the weak topology on the cell complex X. The cell-complexes built by adding cells a dimension at a time are CWcomplexes. Most familiar spaces in homotopy theory are CW-complexes. For example, Sn can be given the structure of a CW-complex in at least two ways; sn = e° U, en, where C: Sn-1 —> e0 is the constant map, and = (e° U e° ) Ui , (e l U e l ) Ui, • • • Ui , (en U en),

where j1 : 53-1 —> Si is the equator and the two j-cells are the upper and lower hemispheres. These decompositions appear in later constructions. If X is a CW-complex, the dimension of X, is defined dim X = sup{n : eal is in the decomposition of X};

94

4. Topological Background

thus dim X < oc. We speak of a finite CW-complex, which is built from finitely many cells, of a countable CW-complex, using countably many cells, and of a locally finite CW-complex, which is built of finitely many cells in each dimension. If X and Y are two CW-complexes, then a mapping f : X —> Y is cellular if f (X (n)) c Y (n ) for all n. We denote by CW, the category of CW-complexes with cellular maps. A first step in establishing this class of spaces as a convenient category for homotopy theory is to show that most of the relevant constructions are functors from this category to itself. We begin by identifying subobjects. A subspace A of a CW-complex X is a subcomplex if A is a CW-complex made up of a subset of cells of X. A CW-complex X with a subcomplex A form a CW-pair (X, A). By taking the cells in X — A together with an appropriate 0-cell for A, one can show that the quotient, X/A, is a CW-complex for a CW-pair (X, A) and furthermore, that X > XIA, the canonical projection, is cellular (see the book of Fritsch -Piccinini90, pp. 62-63]). An example of a CW-complex built via quotients is real projective n-space, RPn that is obtained from Sn by identifying antipodal cells in the decomposition Sn = 0 j, ( e n u e n \) thus IRPn = e ° (e°Ue°)Ui,(el u e l) ui, e l U q2 • • Ll en . For two CW-complexes, X and Y, there is an evident cell-decomposition for X x Y and, if the topology is correct, we have a CW-complex. This works for countable CW-complexes or when one factor is locally finite. A counterexample, due to [Dowker521, shows that it does not work in general: Let X denote the wedge of uncountably many copies of the unit interval with basepoint 0 and form X x X. From the product and quotient constructions we can form the cone on a CWcomplex, CX = X x IIX x {0}, and the suspension, EX = CX/X x {1}. Both constructions give CW-complexes. For a given cell-decomposition of a CW-complex, it is evident that many continuous functions to and from the complex may not be cellular. As in the classical case of polyhedra, cellular maps are sufficient, up to homotopy. —

.

Theorem 4.2 ((he cellular approximation theorem). Let X and Y be CWcomplexes and f: X —> Y a continuous function. Then there is a cellular map g: X

—

> Y with f homotopic to g.

A thorough proof of this theorem for the more general case of mappings of pairs can be found in the book of [Fritsch-Piccinini90, §2.41. We introduce one more level of generality into the discussion and consider the collection of spaces of the homotopy type of a CW-complex. Let W denote the category of such spaces with continuous functions as morphisms. [Milnor591 studied the operation of forming function spaces out of spaces in W. Theorem 4.3. Suppose X is a space of the homotopy type of a (countable) CW-complex and xo G X. Suppose K is a compact metric space. Then the

4.1. CW-cornplexes

95

space map(K, X) = { f : K X f is continuous}, endowed with the compact-open topology, is of the homotopy type of a (countable) CW-complex. If ko G K, then map((K, ko), (X, x0)), the subspace of pointed maps, is of the homotopy type of a (countable) CW-complex. An immediate corollary of the theorem is that, if X is in W, then

= {À: [0,1]

X

A is continuous and A(0) = xo =

the space of based loops on X, is also in W. Restricting to )'Vo, the category of spaces of the homotopy type of a countable CW-complex with continuous functions, we see that Wo is closed under products, subobjects, quotients, suspension and loops. [Milnor631proved that every separable manifold is an object in Wo by a Morse theory argument. Having introduced the category W of spaces of the homotopy type of a CW-complex, we now turn to the homotopy-theoretic properties of these spaces.

Theorem 4.4. Suppose X is a space and f ,g: Sn -1 —> X are two continuous mappings. If f is homotopic to g, then X U1 en is homotopy-equivalent to X U g en . PROOF: Suppose H: Sn' x I —> X is a homotopy from f to g. Define a mapping k : X Ug en —> X Uf en as follows:

k(x) = x for x c X, 1 2i-17 k(t17) =

H(t1,2t — 1)

for 0 < t < 1/2, 17 c Sn" , for 1/2 < t < 1, /7

Reversing the homotopy, one can construct the inverse to k and so k is a homotopy equivalence. This theorem points out the role of the combinatorial structure of a CWcomplex in the homotopy classification problem. In the construction of a finite CW-complex, the n-skeleton, X (n) , is built from the (n — 1)-skeleton by a series of adjunctions. Suppose we have X(n -1 ) and the choice of mappings, frii , fn2, , fnr. : —> X (n -1) . The classes [fn i], , [fnr ] E (X (n-1) determine X (n) up to homotopy. It follows that the computation of 7rn _ i (X ( n -1) ), a skeleton at a time, is sufficient to solve the homotopy classification problem for Wo. The difficulty of this computation is a recurring theme in later chapters. The next theorem, proved by [Whitehead, JHC491, shows how effective the homotopy groups of a space are in distinguishing the space from others. )

96

4. Topological Background

Theorem 4.5 ((he Whitehead theorem). Suppose f: X —> Y is a mapping of path-connected spaces of the homotopy type of locally finite CW-complexes. If 7ri (f): 71-i (X) —> (Y) is an isomorphism of graded groups for i < n and an epimorphism for i = n, then H ( f): Hi(X) —> Ili(Y) is an isomorphism for i < n and an epimorphism for i = n. If 7(f) is an isomorphism of graded groups for all i, then f is a homotopy equivalence. If X and Y are simply-connected and Hi (f): Hi (X) —> (Y) is an isomorphism for i < n and an epimorphism for i = n, then the same holds for 7Ti ( f). If X and Y are simply-connected and H.(f) is an isomorphism of graded abelian groups, then f is a homotopy equivalence.

For arbitrary spaces, such a statement is false. For example, compare a space X with its plus construction X+ with respect to a perfect normal subgroup N xi (X, x0). The associated mapping X —> X+ induces an isomorphism of homology groups, but xi (X, x0) 7r1 (X, x0)/N. (See chapter 2 of the book by [Srinivas96] for the relevant definitions and properties.) We note, furthermore, that it does not follow from an abstract isomorphism of graded groups, 7r* (X) 7r* (Y) or H* (X) H.(Y) that X and Y are homotopy equivalent. When X and Y are CW-complexes, the isomorphism must be induced by a continuous mapping. The machinery of obstruction theory may be employed to build continuous functions from algebraic data when possible. There is a thorough discussion of obstruction theory in the books of [Baues771 and [Whitehead, GW781. In the next section we continue the study of the homotopy theory of CWcomplexes in the more general setting of the homotopy properties of cofibrations. Cofibrations

In constructions leading to spectral sequences, the exactness of certain functors implies relations between the invariants that are expressed in the spectral sequence. For homotopy theory, it is the exactness properties of the functors [—, Z] and [Z, —] that play the principal role. CW-complexes enjoy some of the most useful exactness properties by virtue of their construction. We begin by identifying a topological property that leads to exactness.

Definition 4.6. Suppose X is a topological space and A is a subspace. Then the inclusion mapping i: A —> X is said to have the homotopy extension property (HEP) with respect to a space Y, if for every mapping f: X —> Y and homotopy G: AxI—>Y such that G(a,0) = f (a) for all a E A, there is a homotopy F:X xI —> Y such that F(x, 0) = f (x) and F(a,t) = G(a,t) for x E X, a E A, t E I. The inclusion i: A —> X is called a cofibration f has the HEP with respect to any space.

4.1. CW-cornplexes

97

Iwo examples of cofibrations are important. The first is given by the mapping cylinder of an arbitrary function. Let f : X —> Y be a given continuous mapping and define _ YUXx /

/f(x) (x, 0) ,

the quotient of the disjoint union of Y and X x I by the relation ' (x, 0) to be identified with f (x): The inclusion j: X I, i(x) = (x, 1) is a cofibration (exercise). By construction, notice that Y, as a subset of If, is a deformation retract. In fact, the composite X If —> Y is simply a factorization of f. Thus every mapping can be factored into a composite of a cofibration followed by a homotopy equivalence. The second example is the mapping cone of a continuous function f: X —>

Y. Define

Mf _ Y 11 CX/ f(x) (x, 1).

By taking the quotient of If by the subspace X, we also obtain Mf = /f /X. In this case, the inclusion Y Mf is a cofibration (exercise). Recall that en. If X is a CW-complex, then the adjunction spaces, 471-1) = U f en = Mf, can be written as mapping cones. Thus, the inclusions of X and X (n) X (n+k) are examples of cofibrations. subcomplexes X (n ) Theorem 4.7. Consider the sequence of mappings of pointed spaces (X, xo) For any space Z, the sequence [X, Z] f , Z] is an exact sequence of pointed sets.

(Y, yo)

(Alf, yo).

f'

[Y, Z]

PRooF: First observe that f* 0 i* = 0 f)* carries a homotopy class of a mapping to the constant class since i o f is the composite X —> I —> If /X. Thus im i* c *) -1 {[c]} where c denotes the constant map. Suppose f* [g] = [c], then g o f c, and so it can be extended over CX. Put this extension Z with i*([g']) = [g]. together with Y to get a mapping g' : Y Uf CX Suppose we extend this sequence further by iterating the construction: X

—>YUf CX

3

(Y U f C X) U, CY.

By identifying the cone on Y with the copy of Y in M» we have, up to homotopy, identified Y to a point. Making this line of argument precise, we find that (Y U f CX) ui CY EXX. Continuing in this manner, we derive the Barratt-Puppe sequence GB arratt551, IPuppe581) X

f —>

Y

j

Mf

I

f

EX

Ei

EY

Ej

ME f E 2 X

13 2 f

E 2 Y —> • •

As a consequence of this sequence and Theorem 4.7, we have proved the ex-

actness result:

4. Topological Background

98

Theorem 4.8. Given a mapping f: (X, xo) —> (Y, yo) and a space Z, there is a long exact sequence of pointed sets [X, Z]

[Y, Z]

[M f, Z]

Ef*

[EX, Z]

[EY,

<— • • • .

Some further structure can be identified on this exact sequence of pointed sets. Let SX denote the reduced suspension of (X, xo); that is, SX = EX/{x0} x = X x // x

{0} U X x {1} U x o /.

It follows that SX X. A pointed mapping of SX to Z then takes the form f (x, t) c Z and so, by fixing x and varying t, we get a loop in Z based at f (xo, t) = zo. This determines a mapping [(SX, *), (Z, zo )] [(X, xo), (AZ, cx0 )], which is a bijection. Categorically speaking, the reduced suspension S and based loops S2 are adjoint functors on the category of spaces and homotopy classes of mappings. Let us consider 52Z more carefully. The multiplication of paths, given by A(2t), * tt)(t) = for A,

0
it(2t — 1), 1/2 < t < 1

OZ, determines a multiplication on OZ, m: 1Z x

Z.

This multiplication, in turn, induces a multiplication on [X ,OZ], m.: [X, SIZ] x [X, 1Z]

[X, S-24.

The multiplication m, on the space level, enjoys associativity, an identity, and inverses up to homotopy and so these properties carry over to m.. The next

result is classical. Proposition 4.9. With the multiplication induced by multiplication of loops, the set [X, OZ] is a group. Furthermore, [X, Z] is an abelian group. A corollary of this proposition is that the long exact sequence of Theorem 4.8 is a long exact sequence of abelian groups after a certain point. To show this, it suffices to observe that the isomorphism, [SX, Z] '=" [X, S2Z], commutes with the mappings in the sequence and that the induced mappings of pointed sets are in fact homomorphisms of groups (exercise). Before leaving our discussion of these constructions, we record an important result about the suspension functor to be proved in later chapters. Given spaces X and Y, there is a mapping, E: [X ,Y] —> [S X, SY], given by E([f]) = [S f] (IFreudenthal371).

99

4.1. CW-cornplexes

Theorem 4.10 (the Freudenthal suspension theorem). If Y is n-connected for n > 2 and X is a CW-complex with dim X = q, then E: [X, Y]

[S X, SY]

is an isomorphism when q < 2n and an epimorphism when q = 2n ± 1.

We present a proof of Theorem 4.10 in Chapter 6 (Theorem 6.12). If we restrict our attention to finite CW-complexes, X, then for dimension reasons [S2X, S2 Y] —> • • • , the sequence of mappings, [X, Y] —> [SX, SY] becomes stable after finitely many iterations. We denote this abelian group by {X, Y}o, the set of stable mappings from X to Y. Taking this notion one step given by further we introduce the graded abelian group of mappings, {X,

{x,

—

sny]

These groups enjoy more structure than the unstable sets [X, Y] and so may be studied as a first approximation to the unstable case. In particular, if X = S° = Y, then {so , s0},, 7r', the stable homotopy groups of spheres, which is the subject of Chapter 9. Cellular homology

We next take up the problem of computing the homology of CW-complexes. The combinatorial structure given by a cell decomposition simplifies the computation. This simplification should convince the reader that the category W of spaces of the homotopy type of a CW-complex and cellular maps is an acceptable place in which to do homotopy theory. Suppose X is a CW-complex. We introduce a chain complex associated to X as follows: Let Cell?, (X) = and define 0'11 : Cell?, (X) —> Celln _ i (X) by taking the homomorphisms from the long exact sequences for the pairs (X(), X(n – ')) and (X (n – ') , X (2) ), spliced together as in the diagram Can (X) = H,,(X (n) , X —1) acell

Celln _i (X) =

4. Topological Background

100

,

a

(X ( n -1 )) —> H_, (X (n — 0 , X (n -2 )) —> Since aoi = 0 in the sequence acelloacell _ (iO3)0(i00)= io(00i)o0 = 0. it follows that lin-2(X (n-2) ), We call (Cell. (X), 9"11 ) the cellular chain complex on X. Because a space X may have many cellular decompositions, it is not immediate that H(Cell* (X), 9Cel1 ) is a homotopy invariant of the space X. Furthermore, because Cell„ (X) is a relative homology group, it is not clear what combinatorial data are carried by the complex. We address these questions by determining more of the structure of this complex. Lemma 4.11. Cell, (X) — the free abelian group generated by the n-cells of the cell decomposition of X. PRooF: Write X(n) = X (n-1) UF (Lia e) = A U B where F denotes the attaching maps taken together, A is the image of the open sets, (1/2)ea" ={ G e n such that 1141 < 1/2} and B is X (n) —{image of the centers of each n-cell}. By the excision axiom for singular homology, we have

Celln (X) = Hn (X (n ) , X (n -1) ) Hn (A, A n B)

e Hn ( e SS=') a

e

111

a

eoZe73-1 for the differential on the cellular

Write acell : ea Zena

chain complex. In this guise, acefi is determined by its value on each basis cell and we can write I en , e n —li e n — 1

cell ( e rc-/,,)

L a

.

■3

The integer [ena , e73 -1 ] is called the incidence number of ena and er:3-1 . For CW-complexes that have enough cells to fit together simply, these incidence numbers have a geometric interpretation. For a proof of the following result, we refer the reader to IX.§7 of the book by IMassey911. Proposition 4.12. Suppose X is a regular CW-complex, that is, the characteristic mappings {f a : (en, Sn -1 ) —> (en„, &na )} satisfy the condition that, for all a, fa : en —> cls(ena ) is a homeomorphism. Suppose further that the mappings fa determine the isomorphisms Cell„ (X) '=' ED Zena ' Then a

—1 Ina, e n:3]

= {

0,

if err' ¢ cls(e),

±1,

if e73-1 c cls(en,).

The isomorphisms determined by the fa provide an orientation to each cell. We illustrate this proposition by computing H(Cell. (s2), a"11 ) when 5 2 has the regular cell decomposition S 2 = ( e0 u Ui , (e l U e l ) Uz2 (e2 U e2 ). Denote

4.1. CW- complexes

101

each cell with a + or — to determine upper or lower hemisphere, respectively. We find acell ( e,

e0

e0

acell ( e i

)

eo

eo+

acell ( e, 2

el

el

acell ( e 2 )

el

e+ 1

-+

"

—

acell ) Z, generated by the class e2+ + e2 ; generated by e°E + e° . Computations like this example can be done for such spaces as RP (n), CP (n) and the lens spaces (see exercises), where the geometric construction provides the cell structure and the incidence numbers. It remains to show that cellular homology is a homotopy invariant of the

It follows that

H2 (Cell. (S2 ),

{0} and Ho

Z,

CW-complex.

Theorem

4.13. For X, a CW-cornplex, its integral singular homology IL. (X) is isomorphic to H(Cell. (X), Ocell).

PROOF: Consider the bigraded exact couple arising from the long exact sequences for the collection of pairs (X (P), (P - ')) with Dp ,, = Hp+,(X (P) ), Ep ,„ = Hp+,(X(P), X(P -1 )) and structure maps given by Hp+ , (x(P - 1-)

Hp+ , (x (P))

li Hp±q(X (P) , X(P-1))

a

> Hp±q_ i (X (P- ') ).

By the proof of Lemma 4.11, it follows that, if q 0, Epi = {0} and is given by 0"11 : Cellp (X) —> Cellp _ i (X). Therefore, d1 : E l 0 —> = EZ0 = H (Cell,, (X), 0'11 ).

It suffices to show that E*2 0 based on the following

H.(X). This follows from a diagram chase

Fact. For r > 1, Hp+,(X (P) ) = {O} and Hp (X (P±r) ) Hp (X).

To prove the first half of the fact, we proceed by induction on p. The case p = 0 certainly holds. Consider the exact sequence for the pair (X 9P), x - 1 )) : {O} = 1-129+7.+1(X (P) , X (P-1) ) —> Hp+,(X (P -1) ) —>

Hp+, (X (P) —> Hp+,(X (P)

X (P-1) ) =

{0 . }

102

4.

Topological

Background

By induction, Hp+, (X (P -1) ) = {0 } when r > 1 and so, by exactness, Hp+7,(X(P)) = {0 } . To prove the second half of the fact, we reindex the same sequence: {0 } = Hp±i (X

X (P+r) )

Hp (X (P±r)) —> Hp (X (P±r±1), X (P±r)) = {01.

This implies that Hp (X (P±r) ) is isomorphic to H2 (X (P±r±1) ) for r > 1, and the isomorphism is induced by the inclusion. Thus we have the sequence of

isomorphisms Hp (x ( p+r ))

induced by the inclusions of each skeleton in the next. Any p-chain in X, by the assumption that X is closure finite, must lie in some finite skeleton. Thus the direct limit of these isomorphisms is 112 (X). With this fact, we have collapsed the exact couple into a series of short exact sequences. Splicing these exact sequences together, we get the diagram:

Hp+,

(X( P + 1 ), x( v ) )

o

id O

> Hp (X (P) )

> Hp (X (P) , X(P-1))

a

I

li 1 Hp (X) 1° 0

Define the homomorphism p: (ker di C Hp (X (P) , X (P-1) )) —> Hp (X) by observing that ker = ker 9 = im j H,(X (P) ) and p(u) = i(v) if j(y) = u. Since j is onto, so is p and so it is sufficient to prove that ker p = im . But, by the diagram, ker p^= ken i = im a Aim a) = im . It follows that = ker di / im di ker di / ker p Hp (X) and so H(Cell, (X), acell) For more details on CW-complexes and their properties, we refer the reader to the books of [Massey911, [Lundell-Weingram69], [Steenrod-Cooke-

Finney671, [Fritsch-Piccinini901 and [May99].

103

4.2. Simplicial sets

4.2 Simplicial sets

The decomposition of a CW-complex into cells allows us to compute the homology of such a space more easily. Homotopy groups, however, remain difficult to compute. In this section we introduce simplicial sets and their relation to topological spaces. The combinatorial structure available in a simplicial set leads to another definition of homotopy groups (due to [Kan581). Though the problem of effective computation is not solved by simplicial sets, this different view of a space (as a simplicial set) leads to new constructions. Simplicial sets are the direct generalization of the structure of a triangulated polyhedron and so have their origins in the work of Euler and the inception of topology. A polyhedron may be described completely by giving its set of vertices and then the collections of vertices determined by the faces in each dimension. The description of a polyhedron doesn't suffer if we simply forget that we are talking about faces. Definition 4.14. A simplicial . . . together with functions

K, s: the

set, K., is a sequence of sets, K„, for n = 0, 1, Kn_ 1 , i = 0, 1,... , n, , n, K+1, j = 0, 1,

face and degeneracy maps, respectively, which satisfy

the simplicial iden-

tities di o di = di_ i o di ,

{s i _ 1 o di , o si =

identity,

si 0 di_i, si o si = si±i o si ,

for i < j, for i <

.ffor i = j, j +1, for i > j +1, for i < j.

The schema of ordering of vertices that describe a polyhedron immediately give the data for a simplicial set; a typical n-simplex can be written , v1, ... ,vn ) with vo
where we omit the ith vertex, and the degeneracy maps are given by S 3 0)o, • • • , 210 = @IO, • • • 1 11 j, Vj • • • 1V71)

where the j th vertex is repeated. With this description, the face maps are dual to the inclusion of the i th face into a standard simplex and the degeneracy maps dual to the collapse of the standard simplex onto its ith face.

104

4. Topological Background

The canonical simplicial set is given by the opposite category of the small category, 0, with objects, [n] = { 0,1, . , n } and morphisms, [n] —> [m], that are order-preserving. Let A = 0 01) ; then di : [n] [n — 1] is the map dual to the map { 0,1, .. . , n — 1 } —> { 0,1, ... , n } that skips i in the range and si : [n] —> [n + 1] is dual to the map that repeats j. The simplicial identities follow directly. The category A allows a different description of simplicial sets and a generalization to other categories. Definition 4.15. A simplicial object in a category C is a covariant fitnctor, K: A —> C. A morphism of simplicial objects in C is a natural transformation, f: K —> L. Let SimpC denote the functor category, CA, of simplicial objects in C.

We write K n = K ([0 and di = K (d i ), si = K(si ) to describe the simplicial object as a sequence of objects and morphisms in C, and a morphism iii SimpC can be described as a sequence of maps, fn : Kn —> L n , that commute with the face and degeneracy maps. Let Ens denote the category of sets and .F,A: Ens Ab, the functor from sets to abelian groups that assigns to each set the free abelian group generated by the set. This can be extended one dimension at a time to a fimctor, SimpEns SimpAb. We define the homology groups of a simplicial set K. using ,TA: For each n, let Cn (K.) = .FA(K n )

and 0=

i=o (-1)',FA(di).

The simplicial identities imply that O o O = O and so we define H(K.) = H(C.(K.), O).

Before we introduce a notion of homotopy for the category SimpEns, we connect some of these ideas to the category of topological spaces. The classical construction, motivated by the structure of polyhedra, is the singular complex of a space X, (introduced by [Lefschetz331 and developed by Milenberg441) Singn (X) = {f :

An —> X f continuous},

where An is the standard n-simplex, An = {(x0,

E

x

0, for all i, and

= 1}.

We denote the inclusion of the ith face by si : An -1 An , defined by si(x0,... ,xn_ i ) = (x0,.. • , xi=1,0, xi, • • • 1Xn-1)• The face maps for the singular complex are given by precomposition with these inclusions, that is, : Singn (X) —> Singn_ I (X) is defined di (f) = foc. We denote the collapse of a simplex onto its ith face by nj: A n+1 —> An, defined by

105

4.2. Simplicial sets

xi_i, xi ± xj+i,xj+2, • • • , xri +i)• The degeneracy maps for Sing. (X) are given by precomposition with the collapse maps, sj (f) = fo The singular complex determines a functor Singe : Top —> SimpEns, where Sing. (g) on a continuous mapping g in Top is given by composition. The singular homology of a space X is defined as a composite of functors Hi(X0, • • • ,x+1) = (x (), • • • ,

Sing

H

Top—SimpEns—>SimpAb —> DGAb—>GAb

where DGAb denotes the category of differential graded abelian groups and GAb the category of graded abelian groups. We note that it is possible to normalize the chain complex C. (X) by setting the subgroup generated by degenerate simplices (those in the image of the functions si) to zero. This does not change the homology (Normalization Theorem in [May671) while focusing attention on the nondegenerate chains. We will use this fact in simplicial arguments. If given an arbitrary simplicial set, what distinguishes it as being in the image of the functor Sing.? The condition that must be satisfied was identified by [Kan581 and this condition plays the key role in defining homotopy groups for simplicial sets. Definition 4.16. A simplicial set K. is said to satisfy the extension condition, if for every collection, xo,x 1 ,. ,x i_ 1 ,x i+1,. . . ,x n+1, in Kn , such that dk (x i ) = (11-1 (x ic)

for k < k

there exists x in K n±i with di(x) = xi for j j. If K. satisfies the extension condition, it is called a Kan complex.

Examples of Kan complexes are polyhedra, where the ordering of the vertices is used to establish the extension property, and Sing. (X), for a topological space X, where the retraction of an n-simplex onto all but one of its faces can be used to construct the desired simplex in the extension condition. In the category SimpEns, there is a unit interval given by A[1]. that may be described as a polyhedron with A [1] n = Cvo, • ,VnH v i = 0 or 1 and vo < vi< • • • < vri 1.

More generally, L[m]. has n-simplices vi , ,v with vi E {O,.... 7n} and vo < v1 < • • • < vn • We define the n-skeleton of a simplicial set K. to be the sub-simplicial set that contains all nondegenerate .31 (x)) simplices of degree < n. Observe that A[1]. is isomorphic to the 1-skeleton of A which is the category A = 0°P treated as a simplicial set, that is, as the inclusion

106

4. Topological Background

functor A c Ens. An element of a simplicial set K. that lies in the n-skeleton can be equivalently described by a simplicial mapping A [n]. K. Given two simplicial sets K. and L. their Cartesian product K. x L. is the simplicial set (K. x L.) n = Kn x

di = d x di and si = si x si .

Definition 4.17. If fg: K. —> L. are simplicial maps, then f is homotopic to g if there is a simplicial map, F: K. x A [1]. —> L., with F(x , (0,0, .. , 0)) = f (x)

and F(x, (1,1, ... ,1)) = g(x)

for all w in K.. If f is homotopic to g, we write f

g.

The homotopy relation on simplicial maps K. —> L. need not be well-behaved; however, we have the following result. Lemma 4.18. The relation on the set of simplicial maps from K. to L., SimpEns(K., L.), is an equivalence relation whenever L. is a Kan complex.

The extension condition is there to overcome a difficulty in establishing the transitivity of We refer the reader to the book of [May67] for a proof of the lemma. We next define the homotopy groups of a Kan complex. Let K. be a Kan complex and xo Ko, a choice of basepoint. Propagate xo to each Kp by choosing as distinguished element so o so o • • • o s0 (x0) cK. p times

The n-sphere Sn may be modeled as the quotient simplicial set A [n] /A [n] (n-1) , that is, A [n]. modulo its (n — 1)-skeleton. The set of mappings of pairs, (K., xo)),

is defined by requiring a simplicial map to take elements in (6,[n] (' ) ), to SO o o so (x0 ) in K. The relation is an equivalence relation on this set when K. is a Kan complex and so we define 77, (K., xo ) = SimpEns((A[n].,

A [On-1) ), (K•, x0))/_

=: [(A [72] • A [n] (n-1) ), (K.,

x0)].

To determine further structure on this set, we give an intrinsic definition y if, for all i, of 7„,(K., x0). Suppose x and y are in Kn . We write x

4.2. Simplicial sets

107

di (x) = di (y), and, for some k, 0 < k < n, there is a w in Kn+ 1 such that dk (w) = x, d k±i (w) = y and d,sk (x) = di (w) = di sk (y) for k j# k +1. The simplicial identities imply that — is an equivalence relation. If we let di (z) = so o • • • o 80(x0), for all il, then we can form = {z E K the set of equivalence classes of K7, under — . Since any n-simplex, z E Kn , is representable as a mapping (A[n]., [n] (ri —1) —> (K., xo), we find that 7r,,, (K., X0) = k-n/''' • In this formulation, there is a binary operation, +, on 7rn (K., x0), given as follows: For a, b in kn , let w E Kn +1 be such that do (w) = b, d2(w) = a and di (w) = so o so o • • • o so (x0) for i > 3. Such a class w exists by the extension condition. Define [a] + [b] =[d1 (w)]. For a proof of the following proposition, see the book of 11May67, p. 12 1 . Proposition 4.19. With + defined as above, irn (K., xo ) is a group for n> 1 and an abelian group, when n > 2.

There are now two ways to associate homotopy groups to a topological space; we have the classical definition 70 (X , xo) = [(.9° ,*), (X, xo)] and we have 7r„ (Sing, (X), xo ) = [(A [n],, A [n] (n-1) ), (Sing, (X), x0 ))]. If the extra structure of simplicial sets is going to be an effective tool to study homotopy theory, these groups ought to be related if not isomorphic. The relation between these groups is provided by a construction introduced by [Milnor57[. Definition 4.20. Let K. be a simplicial set and define the geometric realization of K. to be x Kn RK. = 11 n>0

the quotient space of the disjoint union of pairs, ( 77, oc), for x E Kn, E An , by the relation (E7(17),x) ft.', (x)), and (nj(f1),x) si(x)). RK. has the quotient topology. This construction provides a functor R: SimpEns —> Top.

By buildinglIK. one dimension at a time we see that RK. is a CW-complex. Like CW-complexes, when forming products, one must be careful. For two simplicial sets, K. and L., R(K. x L.) is homeomorphic to RK. x if K. and L. are both countable or if one of the spaces, RK. or RL., is locally finite, that is, every point is in the interior of a finite subcomplex. In particular, RAW is homeomorphic to I = [0,1] and so is locally finite. It follows that a simplicial homotopy F: K. x 4[1]. —> L. gives a topological homotopy RF: RK. x I —> R.L.. Furthermore, R induces a mapping [K., L.] —> for L. a Kan complex. In the opposite direction, a topological homotopy, G:XxI—> Y, gives a simplicial homotopy as the composite

Sing. (X) x z[1]. —> Singe (X) x Singe (I) —> Sing, (X x

> Sing, (Y). Singe (G)

108

4. Topological Background

Thus, application of the singular complex functor, Sing., induces a mapping [X, Y] [Sing. (X), Sing. (Y)]. The relation between the functors Sing. and R is described in the following theorem of Kan58'I. Theorem 4.21. Sing. and R are adjoint functors, that is, there is a natural bijection of sets

SimpEns(K., Sing. (X)) <

> Top(RK., X);

moreover, this bijection preserves homotopy.

PROOF: If f : Sing. (X) is a simplicial mapping, then define 4J( f) in Top(RK., X) by Ilf(f )(7, x) = f (x)(21). If g: R.K. —> X is a continuous mapping, let 4)(g) : K. —> Sing. (X) be the mapping, 4)(g) (x)(17) = g(7.1, Clearly, 4f o = id and 4) o = id, giving the one-one correspondence. That these mappings preserve homotopies follows from an argument similar to the discussion before the theorem.

An immediate consequence of this theorem is that our definitions for the homotopy groups of a space X coincide; 7r0 (X, X0) 70 (Sing. (X), oco)• Another consequence is the classical theorem of IHurewicz35/36]. Theorem 4.22 (the Hurewicz theorem). If K. is a Kan complex, then there is

a homomorphism, for m > hm :

7m (K,) -> Hm (K.).

If K. is (n — *connected for n> 2, then ltri is an isomorphism.

SKETCH OF PROOF: Consider the mapping K, .TA(K,), taking each generator in K, to itself in the free abelian group on those generators. This is a simplicial mapping and furthermore, the group property of .T.4(K.) allows one to show that .TA(K,) is a Kan complex. Thus we get a mapping 7r. (K.) —> x (FA(K ,)) . We follow an argument of [Moore561 that H,,(K,) by using the commutative multiplication and the intrinsic definition of 7r.. The second half of the theorem follows by replacing K, with a simplicial set having one simplex in each dimension i, i < n, and which is homotopyequivalent to K,. For such a simplicial set, the result follows directly from the definitions. The classical Hurewicz theorem follows by taking K, = Sing. (X) for X a topological space. We close this section by observing how faithfully the homotopy theory of simplicial sets represents the homotopy theory of spaces of the homotopy type of a CW-complex. As in the case of CW-complexes, an approximation theorem is needed for mappings.

4.3. Fibrations

109

Theorem 4.23 (the simplicial approximation theorem). Given a continuous function, f: BK. RL., where K. is a simplicial set and L, is a Kan complex, there is a simplicial mapping g: K, —> L. with Rg f. Corollary 4.24. The functors R and Sing, establish an equivalence between the homotopy category of Kan complexes and the homotopy category of spaces of the homotopy type of a CW-complexes. In particular, 13 induces a one-one [RK, Rid Top for K. a simplicial set and correspondence, [K., S impE ns L. a Kan complex. Sing, induces a one-one correspondence, [X,Y]Top [Sing, (X), Sing ( Y)l, SimpEns for X of the homotopy type of a CW-complex Y a topological space. and

These results establish the equivalence of homotopy theory in the category SimpEns with homotopy theory in W the category of spaces of the homotopy type of a CW-complex. We exploit this fact in later sections when a simplicial construction may be done more transparently than a topological one. For proofs of these results and the deeper developments of the ideas of [Kan581, we refer the reader to the book-length treatments of [May67], [Quillen671, [André671, [Lamotke68], [Curtis 71] , [Bousfield-Kan72], and [Goers s -Jardine991. 4.3 Fibrations

The construction of a CW-complex involves many natural and desirable features. The building blocks are simple, the operation of adjunction has nice exactness properties, and the homology of the resulting space is calculable from these geometric data. To understand further the importance of the homotopy groups of a space, we give another way to construct topological spaces that enjoys these exactness properties with respect to the homotopy groups functor. We first consider the dual of the homotopy extension property. This property is dual in the sense that it implies exactness on the application of [Z, : Definition 4.25. A mapping, p: E B, has the homotopy lifting property (HLP), with respect to a space Y if, given a homotopy G: Y B and a mapping g: Y x {0} —> E such that p o g(y, 0) = G ( , 0), then there is a homotopy : Y x I —> E such that 0- (y, 0) = g(y , 0) and p 0 = G. Y x {0}

E

YxI

B

A mapping with the HLP with respect to all spaces is called a Hurewicz fibration. A mapping with the HLP with respect to all n-cells is called a Serre libration. By a libration we mean a Hurewicz fibration.

110

4. Topological Background

The property of being a Serre fibration is strictly weaker than that of being a Hurewicz fibration ([Brown, R66]). The properties of CW-complexes can be used to show that a Serre fibration has the HLP for all finite CW-complexes. If p: E B is a fibration, then we refer to the space B as the base space and E as the total space of the fibration. If b is a point in B, then we refer to F5 = p'(b) as the fibre of p over b. Though F5 may vary for different choices of b, the homotopy lifting property restricts the homotopy type of F5. Proposition 4.26. Suppose p: E

B is a fibration and that B is pathconnected. Then, for bo, b1 E B, F50 is homotopy-equivalent to F51 .

PROOF: We prove a bit more here that can be applied later. Given a fibration, B let WB denote the free path space, map( [0, 1], B) = {A: [0, 1] A is continuous}, endowed with the compact-open topology. The evaluation mapping evo : WB B, given by evo (A) = A(0), is continuous. Let Up = {(A, e) E WB x E A(0) = p(e)}

denote the pullback of p: E B over evo : Up Pr2 E Pri

WB evo

The homotopy H: topy lifting problem:

B.

B given by H((A, e), t) = A(t) poses the homoUp (—,0)

Up X I

H B.

B is a fibration, we get a solution fi : Up x I —> E. Let When p: E A: WE denote the adjoint of k given by A(A, e)(t) = i1(0, t). This mapping satisfies the properties p o A (A , e) = A,

and A(A, e)(0) = e.

A is called a lifting function for p. Lifting functions were introduced by [Hurewicz551 and developed by [Fadell-Hurewicz58] and [Brown, E591. The universal properties of pullbacks lead to an equivalence between the HLP for a given fibration and the existence of a lifting function satisfying A(c:p(e) , e)= c, where c, is the constant path at e and A(ct * e) = AO, A(a, e)(1)) ([Fade11601).

4.3. Fibrations

111

Suppose a: I —> B is a path with a(0) = bo and a (1) = b 1 . Consider the composite A evi 4> a : Fbo —> Up —> WE E, where the first mapping is x 1—> (a, x). Since p o A(a, x) = a, A(a, x) (1) G Fbi . Thus the composite x 1—> A(a, x)(1) determines a continuous mapping Fbo —> Fbi . The adjoint of the composite Fbo —> WE gives a homotopy h: Fbo x I —> E between the inclusion of Fbo and 4), : Fbo —> Fbi E. Reverse the path to obtain the homotopy inverse of the mapping. With Proposition 4.26, we can speak of the fibre of a fibration over a pathconnected base space as a representative of the homotopy type of any fibre. The lifting function provides some further structure. Let b E B and let StB = (B ,b) denote the loops in B based at b and let F = Fb. Then 1B >< F C Up and the mapping ,tt = ev i oA: S2B> F. Let a -1 (t) = a(1 — t). Proposition 4.27. To the action p = evi oA: S2B x F —> F and a loop a G SIB associate the mapping ha = tt(a-1 , —): F —> F. Then (1) If oe 0, then h a ho. (2) If a is homotopic to a constant map, then ha idF. (3) If a *0 denotes the loop multiplication of a and /3, then h a * o

ha () ho.

PROOF: (1) Suppose K : I x I —> B is a homotopy with K(8,0) = = (s), and K(0, t) = b = K (1, t). The adjoint of K, k: I -> —> F lands in B. Thus (x, t) 1—> ,u(k(t), x) is a homotopy F between ha and ho. (2)First notice that if c = a constant map, then A(c, x)(1) = x since the lifting problem is solved by constant maps. Now apply (1). , x)(1) = (3)Since (a *O)' = j3 4 * a', the definitions give that A((a A(ct -1 , A(/3 -1 , x)(1))(1). Thus ha* o ha 0 ho. K (s WB

Corollary 4.28. Let G denote an abelian group. If p: E B is a fibration, b G B, a path-connected space, then there is an action of the ftindamental group 7ri (B ,b) on H0 (F; G) and on H* (F; G) induced by [a] ha* and respectively.

Lifting functions and their associated fundamental group action will be developed in later chapters. Examples: (a) Let X be a path-cointected space and xo a basepoint in X. Define the space of based paths in X as the subspace of map([0, 1], X) PX=

{A: [0,1] —> X

is continuous and A(0) = xol.

4. Topological Background

112

The evaluation mapping p: PX —> X, given by p(A) = A(1), is continuous. We show this mapping is a fibration by giving an explicit lifting of a given homotopy: Suppose g:Y —> PX and G: Y >< I —> X are mappings with G(y,0)=p(g(y)), then define 0- :Y x I —> PX by g(y)(s(t +1)), G(y,s(t +1) — 1),

G (,$) y t)( =

for 0 < s < 1/(t + 1), for 1/(t ± I) < s < 1.

This establishes the HLP for p. The fibre over xo is the set of mappings, : [0,1] —> X with A(0) = xo = A(1), that is, the space of based loops X. For a pair of spaces, B and F, the projection mapping, p: B x F —> (b) B, is called the trivial fibration with base B and fibre F. A morphism of librations is a pair of mappings (I, f): (E', B') —> (E, B) such that the following diagram commutes:

B'

with p and p' fibrations. By restricting p : E A c B we get a morphism of fibrations

.13 to the subspace p' (A) for

p-1 (A)

B

A

A fibration is called locally trivial if there is a covering of B by open sets {Va} a G j and a set of homeommphisms { : V, x F —> p-1 (Va)}, E J, each of which induces a morphism of fibrations Va X F Pr i

l

vOE

ç'=" p-1(1/a) C

E

IP

vc

,

c

>B.

It is a slight extension of the proof of Proposition 4.26 to show that a fibration with base space a CW-complex is locally trivial up to homotopy. The main piece of the argument is a theorem of [Feldbau391 that a fibration over a contractible base is trivial. We can then cover the base space by the interiors of n-cells which are contractible.

4.3. Fibrations

113

(c)An example of a locally trivial fibration is constructed from a manifold, M, and its tangent bundle, p: TM —> M. The atlas of coordinate charts provides the covering by subspaces, each of which has the trivial fibration over it with fibre R'n where m = dim M. In this case, the fibration enjoys a great deal more structure than described here. For some of this further structure, see §6.2. (d)From the definition of a covering space, f: E —> X, the covering map is a fibration. The definition of a fibration can be thought of as a generalization of the lifting properties of a covering map. For a history of the notion of fibration see the article of IZisman991. (e)Let G be a Lie group and H a closed subgroup of G. By taking small open sets in G and using the group multiplication, one can show that the canonical projection, G —> G/H, onto the coset space, is a fibration with fibre H. See the classic book of [Steenrod5 I] for details of a proof. These fibrations are studied in Chapters 5, 7 and 8. (f) Consider the classical division algebras over the real numbers: C, the reals; C, the complexes; C, the quaternions; and 0, the Cayley numbers. If we denote any of these algebras by A and d = dimn A, then we define the projective line over A, AP(1), to be the set of lines through the origin in A x A. There is a mapping s2d-1 AP(1), given by sending a unit vector in A x A to the line it spans. Note that AP (1) is the one-point compactification of A and so is homeomorphic to S d . These mappings are called the Hopf librations and are denoted by j: —> n: S3 —> S 2 , v: 57 —> 5 4 , and a: S' 5 —> S 8 . Their fibres are S° , S I- , S3 and S 7 , respectively. Properties of the Hopf librations are considered in Chapters 5 and 9. (g) Suppose A —> X —> Y is a cofibration sequence and Z is a connected finite CW-complex. Then map(X, Z) —> map(A, Z) is a fibration with fibre map(Y, Z). To prove this we check the homotopy lifting property. Let W denote a space. Suppose we have a commutative diagram as on the left: W x {O}

map(X, Z) map(c,Z)

A

X x/

map(W, Z)

Ax/ Using adjoints we can rewrite the lifting problem as a homotopy extension problem as in the diagram on the right. Here the maps are given by f : x 1—> (w (w)(x)) and if: (a,t) (w H (w,t)(a)). Since A —> X is a cofibration, k exists extending f and H. Let :Eh -1/17 x I —> map(X, Z) denote the adjoint of K. Then H solves the lifting problem for the data (f, H). Given the cofibration sequence A —> X —> Y, we may take Y to be X / A. Fixing g: A —> Z, the subset of mappings :9' : X —> Z that extend g is given by extending g away from A, and meeting g at A. This Wx/

map(A, Z)

114

4. Topological Background

describes a map in map(X/A, Z) and determines the fibre of the fibration map(C, Z): map(X, Z)

map(A, Z).

When a pointed space (X, x0) satisfies the condition that the inclusion {x0} X is a cofibration, we say that X is cofibrant. It follows that the evaluation mapping ev,0 : map(X, Z) —> Z,

f f(x 0 )

is a fibration. For example, the evaluation of free paths at any t E [0, 1] gives a fibration evt : WX > X. In order to construct further examples of fibrations, we record some elementary constructions that can be applied to fibrations. The composition of two fibrations is a fibration (exercise). For (X, xo) of the homotopy type of a countable CW-complex and p: E > /3 a fibration, —

—

po

—

: map((X, xo), (E, eo))

> map((X, x0), (B,P(eo)))

—

is a fibration. Suppose p: E B is a Oration and f : X —> .73 a continuous mapping. We can form the pullback of p over f by letting Ef denote the set {(x, e) E X x E such that f (x) = p(e)}. The projection mappings on kif give the diagram Ef

E

Pf

X

B.

The universal property of a pullback has as input data mappings u: Z —> X and V : Z .E; such that f ou = po v and associates to them a unique mapping w: Z Ef with all triangles and squares in the following diagram commutative:

The homotopy lifting property follows from the universal property of pullbacks and the fact that p is a fibration. The fibre of pf : Ef -> X is the same space as the fibre of p. We can use the pullback operation further to show that any mapping can be factored as a composite of a homotopy equivalence and a fibration (compare this

115

4.3. Fibrations

with the dual situation for cofibrations). The evaluation mapping evo : WY —> Y is a fibration with fibre Y. If we are given a mapping f: X —> Y, then form the pullback diagram Uf pT

pri

WY

21,

evo

Y.

X

We show that the mapping 7r = ev i opr i : Y is a fibration, that (If X and finally, that the homotopy equivalence X Uf followed by 7 is the mapping f. Consider the mapping X —> Uf given by 6(x) = (cf (x ), x), where cf( x ) denotes the constant path at f (x) E Y. The composite pr2 0 6 is the identity on X. Let H: (If x I —> (If denote the homotopy given by H((À, x), s) = ((t 1—> A(st)), x). At s = 0 we have H((A, x), 0) = (No), x) = (cf (x ), x) = 6(x) and at s = 1 we have the identity mapping on Uf. Thus Uf X. Furthermore, the composite 7 0 6 equals f. It remains to show that 7 is a fibration. Let Z be a space together with the lifting problem Z

g > Uf

>Y

ZxI

Following [Whitehead, GW781, we introduce the mapping G: G: Z x (al

IUI {0})—>Y

f((Pr2 °9)(z)) = (Pr' ° (z)( 0 ) G(z, s,t) = {G(z,t) (pri o g)(z)(s)

s = 0, s = 1, t = O.

Since (01 x / u / x {0 } ) c / x / is a cofibration (exercise), we can extend G to a mapping on / x I. The adjoint of the extension G:ZxIx1 - -> Y with respect to the middle factor is a mapping a: Z x I —> WY, given by a(z, t) = (s 1—> G(z, s, t)). The homotopy that solves the lifting problem is given by 0(z, t) = ((s 1—> a(z, s,t))>Pr2 o g(z)), and so 7r is a fibration. Thus f can be factored as a composite of a homotopy equivalence and a fibration

We next consider the exactness properties of fibrations.

116

4.

Topological Background

Theorem 4.29. Let E —> B be a fibration and bo a basepoint in B. If Z is a space, then for F = p-1 (bo ) [Z , F]

[Z , E]

[Z , B]

is an exact sequence of pointed sets. The same conclusion holds if p is a Serre fibration and Z has the homotopy type of a finite CW-complex.

PROOF: Suppose g: Z E is a mapping with p o g cbo . Suppose G: Z x B is a homotopy of p o g to the constant mapping. Then we have the diagram Z x {0} E inc

ZxI

G

>B.

By the homotopy lifting property, the homotopy G exists. Let f = zxfiy Since G is a homotopy to the constant map, f determines a mapping into F = p' (b0 ). Furthermore, G is a homotopy from f : Z F E to g and so [g] is in the image of i.. We can extend a fibration to a sequence of fibrations that is dual to the Barratt-Puppe sequence. The key to the construction is the fact that the space P X of based paths is contractible. We proceed as follows: Form the pullback of the path-loop fibration over B with respect to the fibration p: E B:

1 PB

FE

)

B.

Because PB is contractible, Ep has the homotopy type of F, the fibre of p. This gives us a fibration up to homotopy S2B F —> E. By definition, Ep = {(A,e, ) A: (1,0) (B,b0), A(1) = p(e)}. The mapping of 12.13 to Ep is given by w (w, eo) where eo is some choice of basepoint for E in the fibre over bo . If c = cbo is the constant loop at 1)0 E B, then the mapping F Ep given by x (e, x) induces the homotopy equivalence. (Use the lifting function to construct the inverse.)

4.3. Fibrations

tion

117

By a similar argument, we can analyze the pullback of the path-loop fibraover E with respect to the mapping pr 2 : E —> E: SIE

SIB

SIE

PE pri

F Ep pri E.

Here

(1,0) —> (E, eo), A: (1,0) —> (B , bo), 77( 1 ) = E' = {(77 , À,e) e, A(1) = p(e)}. The space E' is homotopy equivalent to SIB by the mapping (n, A, e) 1—> À * (p o 77), where * is loop multiplication. The mapping of SIE to E' is given by Co 1—> (Co, c, e 0 ) and continuing through to SIB we get c * (p )) p . Thus the fibration up to homotopy lE —> SIB —> F has SIp as the 'inclusion' of the fibre SIE in SIB.

We iterate this procedure. Theorem 4.30. Given a fibration p: E —> B with B path-connected and fibre F, there is a sequence of fibrations up to homotopy

onF

onE

onB

Corollary 4.31. For a Serre fibration, F there is a long exact sequence, 7rn

(F)

7rn

7n (B)

(E)

on-1F i p —> ÇIB —> F —> E —> B.

Stnp

E —p B with B path-connected,

7rn _ 1 (F)

—>

P.

7ro (E)

P.

PROOF: Apply [8 0 , —] to the sequence in Theorem 4.30. Then Theorem 4.29 and the isomorphism, [X, SIY] [SX, Y], imply the result.

Some immediate consequences of this corollary are well-known: If j-C is the universal covering space of a space X with discrete fundamental group, then 7rn (k) is isomorphic to 7rn (X) for n > 2. This corollary also implies that 7r, (X) is isomorphic to r_ 1 (X) since P X is contractible. The long exact sequence of Corollary 4.31 shows how a fibration is a sort of exact sequence in Top, up to homotopy. In order to turn more algebra into homotopy theory, we use spaces whose homotopy groups are algebraically determined. These spaces and some resulting constructions are considered in the next section.

118

4. Topological Background

Eilenberg-Mac Lane spaces and Postnikov towers The homotopy lifting property is dual to the homotopy extension property and developing the consequences of HLP leads to a sequence of fibrations that is dual to the Barratt-Puppe sequence of cofibrations. To extend this duality further, we recast one of the principal features of CW-complexes, namely, they are spaces that are built by successive cofibrations. The duality discussed in this chapter was recognized first by Mckmann-Hilton58]. Suppose X is a CW-complex with a fixed cell-decomposition. The sequence of skeleta, X (° ) c X (1) c c X (n ) c • c X, has the following

key properties: (1) X(i) c X (j ) is a cofibration, for j < j. (2) By the proof of Theorem 4.13,

f H(X), {0} ,

if * < n, if * > n.

(3) For a regular CW-complex, the quotient X (n ) I X (n — ') has the homotopy type of a bouquet of n-spheres, V c, S. Furthermore, this sequence of quotients determines H* (X) by the cellular chain complex given by the

cell decomposition. We can summarize these observations in the diagram of cofibrations and quotients: x(n-1)

V al s

C x(n) C

X

e

In this section, we develop a decomposition of a space X using fibrations, a dual of the cellular presentation. This new decomposition is based on the homotopy groups of a space rather than the homology. We begin by dualizing the building blocks in a cell decomposition. The spheres that appear as quotients in the sequence of inclusions of skeleta are distinguished by homology as spaces with one nontrivial reduced homology group in the dimension determined by the sphere. A suitable analogue for homotopy is a space whose homotopy groups are concentrated in one dimension.

Definition 4.32. Suppose G is an abelian group and n > 0 or G is a group and n = 0 or 1. If X is a space such that 7ri (X)=

G, {0},

if j = n, if j n,

then X is called an Eilenberg-Mac Lane space of iype (G, n).

119

4.3. Fibrations

Theorem 4.33. (1) If G is an abelian group and n > 0 or G is a group and n = 0 or 1, then there exists an Eilenberg-Mac Lane space of type (G, n). (2) If two spaces are Eilenberg-Mac Lane spaces of type (G, n), then they have the same homotopy type. For any such space X, we write X K (G,n). (3) If cp: G —> H is a homomorphism of groups, then there is a continuous mapping f: K (G , n) —> K (H , n) with 7rn (f) = (p.

SKETCH OF PROOF: For simplicity, let n > 0 and G be an abelian group. (The general case is similar.) Resolve G by free abelian groups in the short exact sequence 0 —>_11—>F—>G—> O. If F has a basis {ai } ic j and R a basis { bk }kcx, then these two groups and the homomorphism, R F, can be realized by a mapping f: \/v kEK Sn V i , J S". Let Yn be the mapping cone of f. By the Hurewicz theorem, G. Proceed next by induction to attach cells in each dimension, k yn+2 = Yn±i U er 3 U • • • U e 3 , and so yn+i = u e na r u • • • u ena + forth, that kill the homotopy group in that dimension less one. The resulting complex, Yco satisfies 7r„, (Yoo ) = G and xi (Y) = {0 } for j n. Thus Yoo K(G,n).

To realize a mapping, v): C —> H, build the mapping of resolutions 0

R

) F

G

)

0

77 1

H O. 0 The mappings Cp and ri can be realized as mappings of bouquets of spheres and so induce a mapping in the construction above. In particular, the identity mapping, G —> G, can be realized for two different complexes that are candidates for K (G, n) by mapping appropriate resolutions. By the Whitehead theorem, these two complexes are homotopy-equivalent.

Some examples of Eilenberg-Mac Lane spaces may be given from the theory of covering spaces. First, we have the fibration R 5 1 , the universal covering space of the circle. By the long exact sequence of homotopy groups for this fibration and the contractibility of R, K(Z, 1). For G = 7,122Z, there is the covering space for each n, given by 2772E Sn RP(n). These fibrations fit together in the system: Z/2E

S2

RP(2)

2Z/2Z

) S3

c RP(3)

120

4. Topological Background

The direct limit is a fibration, Z/227_, S' —> RP(oc) with S weakly contractible (that is, 7ri (S') = {0} for all i). The long exact sequence of homotopy groups determines the homotopy groups of RP() and so RP (co) K(712E, 1). For the complex projective spaces, we have a similar system of fibrations 5292+1 cp(n ) The direct limit is a fibration with fibre Sl; 51 S"

CP(oo)

and, since 5' K(, 1), the long exact sequence of homotopy groups implies that (CP(oo) K (E, 2). Finally, if given a K(G, n), the path-loop fibration PK(G,n) K(G,n)

implies that K(G, n) K(G,n — 1). We remark that the sequence of spaces, K(G,0), K(G,1), , K(G,n), . . . comprise an Omega-spectrum called the Eilenberg-Mac Lane spectrum for G. Spectra and generalized homology and cohomology theories are not developed in this book (see, for example, the recent book by IKochman961). It is useful to recall the following classical result. The proof of this theorem may be given by an obstruction theory argument (IrEilenberg47]). Theorem 4.34. Let G be an abelian group. Then there is a natural isomorphism, for each n and each CW-complex Y, Hn (Y; G) [Y, K (G,n)].

This equivalence led to the definition of the generalized cohomology theories and plays a crucial role in many arguments in homotopy theory. With the Eilenberg-Mac Lane spaces for building blocks, we next introduce the Postnikov system of a space X (IPostnikov511), which provides a decomposition of X that is dual to the cellular decomposition. Theorem 4.35. To a simply-connected space X, there is a sequence of spaces, {P,,X for n = 0, 1, ... } and sequences of mappings,

= 2 ,... {132 : PiX —> Pi _iX { /7 : X —> /33 X 1 j = 0,1,... } {km : Pm _ i X —> K(7,(X), m 1) m = 1, 2, .. } such that the following hold:

(1) each mapping, A, is afibration with fibre K (7,(X),i), pulled back over Pi _ i X from the path-loop fibration with respect to the mapping le: P,X

PK (7,(X),i 1)

13

K (7,(X),i +1).

4.3. Fibrations

121

(2) For each space, P,X, 7,n (1, X)

f 717n(X)' {0},

for m < j, for m > j,

and the mapping fi: X Pi X induces the isomorphism —> (pi X) for m < j. (3) For all j , h = pi+1 0 fi±i.

f: 7rm (X)

Such a system of spaces and maps is called a Postnikov system for X, and the data contained in the system determine X up to homotopy.

P2X

K(73 (X), 4)

11)2

Pi X

k-

K(72

(X),

3)

lpi PoX

In the case that X is not simply-connected, the tower of fibrations {Pi X, pi } may be constructed with the fibre of pi given by K(.7ri (X), j). However, the existence of the family of mappings {km }, called k-invariants of X, make each of the fibrations pi a principal fibration. This would force the fundamental group ri (X) to act trivially on the higher homotopy groups of X, which is not true in general (see Chapter 8 bis for more details). SKETCH OF PROOF: Here is an instance where working simplicially allows us to make a very straightforward definition that would be cumbersome to make with spaces. Replace X with a Kan complex L. Consider the following equivalence relation, defined for each n: Two q-simplices in L., u and y, satisfy u y if each face of u of dimension < n agrees with each face of y of dimension < n. Equivalently, if xu : [q]. —> L. and xv : [q], L. are the characteristic mappings for n and v, then xu and xv agree on the n-skeleton [q].. Let (L.),„ = L./ -,„„. [q] It follows from the definitions in simplicial theory that each (L.),, is a Kan complex, that 7,(L.) = 7r3 ((L.)), for i < n and 7r((L.)„) = {0} for i > n, that (L.),,, (L.)_ 1 is a simplicial fibration with fibre a K(71-,, (L.), n) (see the article of ICurtis711 for details). To complete the proof for spaces, let L. = Sing, (X) and then apply the realization functor to the subsequent construction.

4. Topological Background

122

The Postnikov system of a space and its generalization for mappings ([Moore58]) are powerful tools in the study of homotopy theory. Some of the computational potential is explored at the beginning of Chapter 6, and in Chapter 8bis ; analogous constructions are considered in Chapter 9. 4.4 Hopf algebras and the Steenrod algebra

We close our consideration of topological preliminaries with a discussion of some of the properties of the singular homology and cohomology functors. The algebraic structures that we consider arise naturally from the functoriality of homology and cohomology and from a choice of basepoint. They have become basic structures in the study of algebraic topology and appear in many guises in the remainder of the book. Fundamental to all subsequent structure is the following equivalence. Theorem 4.36 (the Eilenberg-Zilber theorem). For any pair of locally finite spaces, X and Y, and any field k, there are natural isomorphisms AW : H * (X Y ; k) <

H* (X; k) H * (Y ; k) : EZ,

AW : H* (X x Y; k) <

If* (X; k) H*(Y ; k) : EZ .

Furthermore, these equivalences are induced by chain equivalences on the singular chain and cochain complexes, respectively.

SKETCH OF PROOF: For complete details, the reader can consult the classic text of [Mac Lane63, Chapter VIII, §81. We record the chain maps for later use: AW: C>, (X x Y; k)

C* (X; k) C * (Y ; k)

is known as the Alexander-Whitney map and it can be described simplicially by n <— AW (a x b) =

dn _ ia 0 di b, i=o

the mapping that sends the product of simplices to the last n — i faces of a (dn _ia) tensor the first i faces of b (di b). We record that on chains AW is not a mapping of differential graded coalgebras since it is not cocommutative. The inverse mapping (up to chain equivalence) is the Eilenberg-Zilber map EZ: Cp (X; k) 0 C q (Y;k)

Cp+ q(X

x Y; k)

described by EZ(a 0 b) =

E

(p,q)—shuffles

(_1)6(0-)(s 0. (1) • • • so. (p) a X 5 a(p +1) • • • S (p+ q)b) ,

123

4.4. Hopf algebras and the Steenrod algebra

where the sum is taken over all (p, q)-shuffles. A permutation of p q letters is a (p, q)-shuffle if o- (1) < • • • < o- (p)

and o- (p+ 1) < • • • < o- +

The mappings si are given by the degeneracy mappings for the underlying simplices and the sign is defined in various ways (see §7.2 for one of them). We note that the Eilenberg-Zilber map on cochains is an algebra mapping. (These ideas come up algebraically in Chapter 7.) Suppose (X, x 0 ) is a pointed space. Then X is immediately equipped with three continuous mappings: A : X —> X x X, the diagonal mapping, A (x) = (x, x); the basepoint map, xo : * —> X, xo(*) = xo, where * is the one-point space and xo is both the basepoint and the mapping identifying it; finally, there is the unique mapping, X —> *, of X onto the terminal object in Top. We apply IL ( ; k) to each of these mappings and obtain homomorphisms A = AW 77

: H* (X ; k) —> H * (X ; H* (X; k),

=(xo): k

and

E:

H* (X ; k), H(X;k)

k.

We refer to A as the comultiplication on H* (X; k), n as a unit on H* (X;k) and E as the augmentation of II* (X; k). We abstract the algebraic structures that result from these choices (following Wilnor-Moore651). Definition 4.37. Let H* denote a graded vector space over afield k. We say that II* has a comultiplication if there is a mapping A : If H 0 H.

(1) A is coassociative if the following diagram commutes A

H*

H* 0 H*

10A H* 0 H* 0 H*.

(2) A has a counit if there is an augmentation of H* ,

E:

following diagram commutes

k

o H*

6(

,,

H* H*

H* 0 k.

H*

k and the

4. Topological Background

124

The triple (Ifs , A, E), satisfying (1) and (2), is called a coalgebra over k. (3) A homomorphism, 77: k II* is a unit ,ffor the coalgebra H* if the composite so77:k k is the identity mapping on k. (A unit may also be called a supplementation of If s .) (4) A coalgebra is said to be cocommutative if the following diagram commutes,

) II 0 Hs

11* 0 II

where T(a o b) = (_ 1 )(deg a)(deg b)b ® a. Proposition 4.38. For a locally finite space X the singular homology of X with coefficients in afield k is a cocommutative coalgebra over k. For a choice of basepoint, H* (X;k) also has a unit. Some familiar algebraic objects enjoy the structure of a coalgebra. For example, let A(x) denote the exterior algebra on a generator x with dim x odd. Equip A (s) with the comultiplication determined by A (x) = 1 x + x 01. This is the unique coalgebra structure on A(x) up to multiplication by scalars. On k[x], the polynomial algebra on x of even dimension, there is a comultiplication given by A (s) = 1 0 x + x 0 1 and

A(xn) = 1

xn

+

1

xn -1 + • • • + (n) x k

Xn k ± • • • ± X n

1.

These examples of coalgebras are realized by 11* (S 2 '; k) and II* (CP ( Do); k). An important subspace of a coalgebra is the subspace of primitives, Prim(Hs ) = {s E IL. A(x) = 1 Ø x + x 0 1}. The importance of the space of primitives becomes apparent when the coalgebra structure is combined with an algebra structure (as in a Hopf algebra). For our examples, Prim(A(x)) = k {x} and Prim(k [x]) = klx , xP , xP 2 , 1, where p = char k, and kS denotes the vector space with basis the set S. If we apply cohomology with coefficients in a field to the diagonal, basepoint and terminal mappings, the Eilenberg-Zilber theorem gives a multiplication, the cup product on cohomology, H* (X; k) 0 H* (X; k)

H* (X; k),

125

4.4. Hopf algebras and the Steenrod algebra

an augmentation, E : H*(X;k) k and a unit, p: k H*(X;k). As defined in Chapter 1, a graded algebra over k is the appropriate abstraction of these notions. By applying the vector space dual to the definition of coalgebra, we can define an algebra as a triple, (H*, cp, 77), with cp, a multiplication that is associative and has a unit, n. Furthermore, the dual of Proposition 4.38 holds, namely, H* (X; k) is a graded commutative algebra over k and, for a choice of basepoint, H* (X; k) has an augmentation. The dual notion to the subspace of primitives of a coalgebra is the space of indecomposables of an algebra H*: Let I(H*) = ker E, then define _ I(H- ) I

P (I (H )

I (H* )) •

For the examples of A(x) and k[x], considered as algebras, Q(A(x))^=- k{x} and Q(k[x]) k{x}. We record some consequences of duality. Theorem 4.39. Let II* denote a coalgebra over k with augmentation E and unit n. Suppose II* is of finite type, that is, dimk ii is finite for all n, then (1)

(Hal, Adual , Edua1 )

is an algebra over k. A dual is commutative. (3) ndual is an augmentation of H1. (2) If i.ï is cocommutative,

Next we consider spaces with some further structure. Suppose that X is an H-space, that is, there is a continuous mapping, m: X x X —> X, a multiplication and furthermore, for a choice of basepoint, e G X, the mappings (x) = m(x, e) and mr (x) = m(e, s) are homotopic to the identity on X. Examples of H-spaces abound: If (G, e) is a topological group, then it is an Hspace. The loop multiplication* on (X, xo) with basepoint cso gives S2(X, xo) the structure of an H-space. Finally, notice that the group multiplication on an abelian group G is a group homomorphism and so induces a mapping, for each n, K (G , n) x K (G , n) K (G , n); this can be seen to be an H-space structure. The homology of an H-space with field coefficients is endowed with a multiplication, m* : H* (X ; k) 0 H * (X ; k) H * (X ; k), called the Pontryagin product. Dually, the cohomology algebra obtains a comultiplication, m*. The properties that result are abstractly given in the following definition. Definition 4.40. Let H denote a graded vector space of finite type over a field k, equipped with a multiplication cp: H H —> H, a comultiplication, :H H H, a unit, 77: k —> H, and an augmentation, s: H —> k. Then H is a Hopf algebra over k if (1) (H, cp, 77) is an algebra over k with augmentation (2) (H, A, E) is a coalgebra over k with unit p.

E.

4. Topological Background

126

(3) The following diagram commutes so

H H

H

›HOH s,e ,p

A0A

H

A

)11- (H

leTel

)HOH0H6DH.

That is, A is a morphism of algebras or, equivalently, is a morphism of coalgebras.

We have assumed in the definition of algebra and coalgebra the conditions of associativity and coassociativity. Without these conditions (H, 0, A, n) is a bialgebra, a structure at the heart of the study of quantum groups OhniderStemberg93]).

Theorem 4.41. If X is a locally finite H-space, then H* (X; k) is a commutative Hopf algebra over k and H.(X; k) is a cocommutative Hopf algebra over k. Furthermore, the Hopf algebras IL(X;k) and H* (X; k) are dual to each other The dual of a Hopf algebra of finite type is also a Hopf algebra. The structure of Hopf algebras, as developed in the study of Lie groups and Hspaces, is well-known in the useful cases (see Theorems 6.37 and Theorem 10.1 for examples). We next present some further structure on singular cohomology—the Steenrod operations. Taken together these operations have the structure of a Hopf algebra over the field of p elements Fp known as the Steenrod algebra. When distinguishing two spaces by their cohomology with field coefficients, the first piece of structure available is that of a graded vector space. When two spaces have isomorphic graded vector spaces for their cohomology (for example, CP(2) and S2 V S4), then the cup product structure on the cohomology may distinguish the spaces by finding different graded algebras. However, this too may fail to distinguish the spaces (for example, ECP(2) and 53 V S5) and so further structure is desirable. Definition 4.42. A cohomology operation of type (G, n, G' ,m) is a natural transformation, 0: Hn ( ; G) —> Hm( ;G'). That is, for all spaces, X and Y, and mappings f: X —> Y, there are functions 0x, 0y, such that the following diagram commutes 9x ) Hn (X; G) (X ; G I )

Hn (Y ; G)

4.4. Hopf algebras and the Steenrod algebra

127

Over a field, k, the cup product on H* (X; k) allows us to define a squaring map 0(u) = u u. This operation 0 is of type (k, n, k, 2n), for each n. Consider the short exact sequence of rings 0 —> z7/p2Z,

z7,/p2 E —> 7Z/p27_, —> O.

If we tensor C* (X), the integral cochains, with this sequence of coefficients, we get a long exact sequence on homology • • • —> Hn (X; 1/p2 1) —> Hn (X; 1/pZ,)

0

Hn+1 (X;11p1) —> • • • .

For each n > 1, there is a connecting homomorphism, natural in X, that determines a sequence of operations, /3, of type (Z/pZ, n, Z/pZ, n 1). These operations are called the Bockstein homomorphisms (see Chapter 10). More generally, one can use Theorem 4.34 to produce cohomology operations. Let 6 be any class in [K (G , n), K (G' , m)] and u, a class in Hn (X; G). Then u = [f] for f: X —> K(G,n) and so composition with 0 determines a class in [X, K (G' , m)] = Hm (X ; G') which is denoted by 0(u). By the same kind of obstruction theory argument that proves Theorem 4.34, one can prove the following characterization of operations. Theorem 4.43. Let Oper(G, n,G',m) denote the set of all cohomology operations of type (G, n, G' ,m). Then there is a one-one correspondence between Oper(G, n,G' , m) and [K (G ,n), K(G' ,m)].

Since [K (G , n), K (G' , in)] Htm (K (G , n); G'), it becomes necessary to compute the cohomology of Eilenberg-Mac Lane spaces in order to determine cohomology operations. This problem is taken up in Chapter 6. In Chapter 9, there is a detailed discussion of the application of these operations in homotopy calculations. The collection of all cohomology operations, for a given field k, is sometimes a large, unmanageable set and so it is unsuitable for straightforward application. We reduce our attention to a more manageable set by considering sequences of operations that are independent of dimension and related by the suspension isomorphism. Definition 4.44. Let {0,, E Oper(k, n,k,n + i)} denote a sequence of cohomology operations for i fixed. Then {0„} determines a stable cohomology operation of degree i if the following diagram commutes

fi n ; k)

E

e, fin*Pi (X; k)

> fin±l (S k) en+1

E

fr±i±l(SX; k)

4.

128

Topological

Background

where E: if' (X ; k) kn+ 1 (S X; k) is the suspension isomorphism, E: n (X ; k) [X , K (k , m)] [X ,QK (k , m 1)] [S X , K (k , m 1)].

An example of such an operation is the Bockstein homomorphism: We can write the suspension isomorphism explicitly as the mapping determined by the diagram if' (X ; k)

ifm+ 1 (S X ; k) N

flm+ 1 (C+ X , X: k) <excision fi m+1 (SX, C_X;k)

where SX = C± X U C_ X, the union of two cones. The Bockstein homomorphism commutes with the mappings of pairs and so with the suspension isomorphism. [Steenrod471 introduced a family of stable cohomology operations over 72. With composition as product, these operations form an algebra, denoted A2 and called the mod 2 Steenrod algebra. The analogous operations were constructed by [Steenrod521 over If, odd primes, p, to form the mod p Steenrod algebra, Ai, . In the early 1950's, [Cartan501, and [Adem521 established the structure of these algebras and then [Serre53] and [Cartan541 showed that Steenrod's constructions gave all possible stable cohomology operations over the prime characteristic fields. Steenrod's constructions take place at the level of cochains and involve certain equivariant acyclic complexes and the cohomology of groups. We refer the reader to the readable accounts of [Steenrod-Epstein621 and [MosherTangora681 for this construction. Alternative constructions of the Steenrod algebra may be found in the papers of [May701 and Karoubi951. The structure and action of the Steenrod algebra is described in the following theorem, which is stated twice—for the prime 2 and for the odd primes. Theorem 4.45 (mod 2). The mod 2 Steenrod algebra, A2, is generated by operations ( ; 72), Sq' H*( ;72) —> If for i > 0, satisfying (1) Sq ° = the identity homomorphism. (2) If x E Hn (X: 72), then Sex = x2 . (3) If x E Hn (X:72) and i > n, then Sq' x = 0. (4) For all x, y E H* (X ;7 2), Sq k (x y) =

2 ,i–o Sq'x Sq k–i y,

the Cartan formula.

129

4.4. Hopf algebras and the Steenrod algebra

(5) Sq l is the Bockstein homomorphism associated to the short exact sequence of coefficients, z/27, x2

Z/LIZ —>—> O.

(6) The following relations hold among the generators: i f O < a < 2b

sq asq b

[a/2]

(b

1 0 -

a

1 Ti) sqa+b-i - 23

These relations are called the Adem relations. Theorem 4.45 (for p an odd prime). The mod p Steenrod algebra, A p, is generated by operations P? : H*( ; Fp) H*+ 29 (P-1 ) ;F), for i > 0, H*( : Fr ) H*±1 ( ; along with the Bockstein homomorphism, associated with the short exact sequence of coefficients, xP 277p2 E —> 7 /pZ —> O. 0 —> E/pE —> These operations satisfy the following (1) (2) (3) (4)

P0 = the identity homomorphism. If x E H 2"(X; Yp ), then P"x = x. If x E Hn (X: Fp ) and k > 2n, then Pk x = O. For all x, y E H* (X; Fr ), the Cartan formula holds

pk (x y)

Pix Pk - iy.

Also, 13 is a derivation,

3(x

1

y) = 13x

y+ (-1) di' x

/(iy.

(5) The following relations hold among the generators: pa pb \ ,[a/P1 ((p - 1 )( 1) i) - 1 ) pa+b—i pi 3.=0

apj

and ifa < pb, then pa/3 pb

[a /

1)a+

j+0 [(a-1)/p]

1)(b_ ))Pa+blPI

3 ((p -a - pj

H1r-1,j-1((P 1)(b - j) -1)

a - pj - 1 These relations are called the Adem relations.

pa+b-3 pj

130

4. Topological Background

In these theorems, all binomial coefficients are taken modulo the prime and [ ] denotes the greatest integer function. An elegant and useful reformulation of the Adem relations has been given by [Bullett-MacDonald821. We can use the mod 2 Steenrod operations to distinguish CP (2) and S3 V S5 . The difference is detected by Sq2 (H3 (ECP(2); F2) {0} = Sq 2 (H3 (S. ' V S 5 ;79). The applications of these operations pervade algebraic topology. For further examples we refer the reader to Chapters 6, 8, 9, and the classic book by [Steenrod-Epstein62].

We close this section by recording some useful facts about the Steenrod algebra that will be applied in later chapters. FACT 1: The vector space of indecomposables of A2, is given by Q(A2)

F2{Sq l ,Sq2 ,

,Sq 2 ' ,

}.

For p, an odd prime, (Ap) = p{0, P 1 , PP

PPk .1.

The space of indecomposables of an algebra generates it by taking the span of all products of indecomposables. FACT 1 follows from the Adem relations. To determine a vector space basis for each Steenrod algebra, we push further with the Adem relations. Theorem 4.46 (mod 2). Let I = (j1, i2,* ,i,r ) be a sequence of nonnegative integers. We say that I is admissible if i s _ i > 2i8 for r > s > 1. Associate to I the product of generators Sql = Sqii Sq i2 • • • Sq. We say that Sq l E A2 is admissible if the sequence I is admissible. The admissible products form a vector space basis for A2.

Theorem 4.46 (for p an odd prime). Let J be a sequence of nonnegative integers, (60, 81 ,6 1 , , sk,ck), where c i = 0 or 1 for all i. We say that J is Ei for k > i > 1. Associate to J the product of generators 0fpJ = 0E0 psi 0E1 . psk i3Ek We say that PJ E Ap is admissible if the sequence J is admissible. The admissible products form a vector space basis for A.

admissible if si > psi+i

FACT 2: Let A : Ap

Ap Ap

be the comultiplication that extends the

Cartan formulas: for p = 2, A (Sq k ) =

Sqi Sq k—i , and for p odd,

AO) = 1 0 + 0 0 1, and A(P k ) =

pj pk-j .

j=0

Then A p is a cocommutative Hopf algebra over IF for any prime, p.

131

Exercises

This fact was first observed by [Milnor581. It follows that H*(X; 7p ) is a module over the Hopf algebra A p for any space X. Furthermore, the dual of A p, Apduai is a commutative Hopf algebra over IF. From the structure theorems for Hopf algebras proved by [Bore1531 and [Milnor-Moore651 and the known action of A p on certain test spaces, [Milnor58] computed the structure of Ad": Theorem 4.47 (mod 2). As an algebra, A" 1 is isomorphic to the polynomial algebra 72

[

6. 1 '21

where deg i = 2 — 1. As a coalgebra, the comultiplication on A ual is c2'

determined by A(k) =

c

Theorem 4.47 (for p an odd prime). As an algebra, A pdual is isomorphic to the tensor product of an exterior and a polynomial algebra, A(To, T1, • • • )

where deg ri = 2pi — 1 and deg plication on A pdual is given by

A(6c) =

19p

, 6, • • •

= 2(pi — 1). As a coalgebra, the comulti-

AN) =

i=

The implications of these structure theorems become apparent in Chapter 9 where they figure prominently in the calculations associated with the Adams spectral sequence.

Exercises 4.1. Suppose that f: X > Y is a continuous function. Show that the inclusion : X —> If, the mapping cylinder, is a cofibration. Show that Y is a deformation retraction of If . —

Show that the composition of cofibrations is a cofibration. Show that the composition of fibrations is a fibration.

4.2.

4.3. Suppose that f: (X, x 0 )

(Y, yo ) is a map of pointed spaces. Show that the

iterated mapping cone sequence f X —> Y —> Mf = Y Uf CX (Y Uf CX)

Ui CY

gives a space (Y Uf CX) U CY homotopy equivalent to X.

132

4. Topological Background

4.4. Let SX denote the reduced suspension on X. Show that for (X, xo) and (Y, ye ) pointed countable CW-complexes there is a isomorphism of groups [(SX, 8 x0), (Y, ye)]

[(X, xe), Q(Y, Yo)]

•

Show further that for a mapping of pointed spaces f: (X, x 0 ) > (Y, ye ), the mappings induced by the iterated mapping cone sequence give an exact sequence of groups: —

Ef*

[EX,

[Y,

Ei*

[EMf ,

Ea*

[E 2 X,

4.5. Prove that the suspension mapping E: [Sk X, Sky] a homomorphism for k > 1.

[Sk+ 1 X,Sk+ 1 11 ] is

4.6. Extend the discussion of cellular homology to include coeffi cients in an abelian group G. Compute H-,.(RIP(n)) and H-,.(R.P(n);72 ) from an appropriate cell decomposition. What decomposition would be appropriate to compute 11,, (CP (n)) cellularly? Let Z/mE act on the circle as roots of unity (a generator of E/mE corresponds to a primitive mth root of unity in 5 1 ). Since 5 1 acts on 52n -1 , the unit vectors in EC, this gives an action of E/mE on S2n -1 . Let L" – ' (m)denote the orbit space of this action; this is an example of a lens space (Whitehead, p. 91]). Compute the cellular homology of L 2 '' (m). GW78, 4.7. Determine explicitly the lifting function for the path-loop fibration over a space X. This leads to an action of S2X on itself. Determine that action explicitly. 4.8. If p: E

> B is a fibration and (X, x 0 ) E V V, then show that precomposition with p gives a fibration: p

—

—:

map( (X, xe), (E, e 0 ))

—

> map( (X, x e ), (B,p(eo)).

4.9. Show that (01xl - U1- x {0}) C I>< / is a cofibration. 4.10. Suppose that A is a connected Hopf algebra over a field k, that is, A = k e 1(A), where 1(A) consists of elements in positive degrees. Show that the space of primitives is the kernel of the reduced comultiplication A : /(A) C A

A A —> /(A) /(A)

where we can take /(A) isomorphic to the cokernel of the unit 77: k that there is a natural homomorphism Prim(A) Q (A).

—

> A. Show

4.11. A sequence 0 K H Q 0 of Hopf algebras is a short exact sequence if K is a sub-Hopf algebra of H and 1(K) • H = H • 1(K), that is, K is a normal sub-Hopf algebra, and Q k OK H HI I(K) • H as graded k-vector spaces. Show that the functor that associates to a Hopf algebra its subspace of primitives is left exact, that is, a short exact sequence of Hopf algebras, 0 K H —> Q 0 gives a short exact sequence of vector spaces 0 Prim(K) —> Prim(H) —> Prim(Q).

5 The Leray-Serre Spectral Sequence I . the notion of a fibre space hides subordinate structures, now revealed only in part by the array of differentials of the associated spectral sequence." S. Mac Lane

After 1930, new invariants of a topological space X were introduced: the higher homotopy groups 7„(X) by [Hurewicz35/361 and [tech321, and the cohomology ring by [Alexander35], [Kolmogoroff361, and [Whitney381. The various homology theories (simplicial, de Rham, tech, singular, etc.) were clarified through the axiomatization of [Eilenberg-Steenrod451. Computations of these invariants proceeded slowly, especially for spaces of importance, such as manifolds, Lie groups, and their associated homogeneous spaces. The inclusion of a closed connected subgroup H of a compact Lie group G leads to a libration (indeed, a fibre bundle) H G —> GIH. Considerable effort was directed at discovering the relations between the combinatorial invariants of H, G, and G/H. [Hurewicz35/361 already appreciated the relations among the new homotopy groups for (H, G, G H). Although he did not have the language of long exact sequences, he had shown what was equivalent to the exactness of the sequence of groups ri (G)

—> 7,,(G I H) —> 7r, _ 1 (H) —>

and he gave the isomorphism r1 (S3 ) 71-i (S2 ) for j > 2 as an example. Other geometric problems, like the existence of nonzero vector fields on manifolds, led to topological structures with properties analogous to the fibre bundle (H, C, G I H). [Whitney351 introduced sphere bundles in this context and developed his theory of characteristic classes toward the classification of sphere bundles. Throughout the 30's and 40's a search for the most general and useful definition of a fibre space occupied many topologists (a history of this search has been written by [Zisman991). The most influential efforts (recounted in the problem list of [Massey551) were due to [Hurewicz-Steenrod411 who introduced slicing functions on metric spaces, to [Ehresmann-Feldbau411 who sought to generalize to topology the geometric notion of connection in a fibre

134

5. The Leray-Serre Spectral Sequence I

bundle with smooth structural group (see [Steenrod52]), and to [Eckmann42] who focused on the homogeneous space case and its homotopy lifting properties. During the Second World War, [Hopf411 introduced H-spaces as a generalization of Lie groups. He also showed that the cohomology of an H-space enjoyed the additional structure of a Hopf algebra. [Samelson411 developed Hopf's ideas further and introduced the notion of a closed subgroup H of a compact Lie group G being totally nonhomologous to zero in order to show that under these circumstances H.(H;Q) 0 H.(G I H;Q) is isomorphic to H(G; Q). Rlopf42] also studied the class of spaces dubbed aspherical by [Hurewicz35/36] for which 7r1 (X) = {0} for j > 1. Hurewicz had shown that the homology groups of X were determined by the fundamental group. Hopf went much further when he gave an explicit formula in terms of the fundamental group, for any space Y, for 112 (Y)/h(72(Y)), where h denotes the Hurewicz map. When specialized to an aspherical space X, this gives the second homology of the group 71 (X). This result launched the theory of the homology of groups (see Chapter 8bis). The notion of Eilenberg-Mac Lane spaces grew out of the generalization of Hopf's work to higher homotopy groups and higher orders of connectivity ([Eilenberg-Mac Lane45]). ]Leray46] solved the problem of relating the cohomology rings of spaces (F, E, it, B) making up a fibre space. He gave the first explicit example of a spectral sequence in the cadre of sheaves and a general cohomology theory that he had formulated while a prisoner-of-war (see [Leray45]). Leray's cohomology ring (based on his couvertures) specializes to de Rham cohomology, to tech to Alexander-Spanier cohomology, and to singular cohomology cohmlgy, with the appropriate choice of complexes (see [Bore151, 98 1 and [Houze1901). Refinements of Leray's algebraic apparatus were due to [Koszu147] and [Cartan48]. Their work adapted well to the study of homogeneous spaces. For the computations of homotopy groups, however, a tool like Leray's spectral sequence was needed for singular homology for which theorems like the Hurewicz theorem and the Whitehead theorem reveal a close computational relation. Such a spectral sequence for singular homology appeared in a series of Comptes Rendus notes by [Serre50]. Complete details appeared in his classic thesis ([Serre511) that introduced the very general notion of a (Serre) fibration making many computational examples possible and even computable. A short time later, [Bore153] published his Paris thesis in which the computational power of spectral sequences was used to extend the known results on the homotopy theory of Lie groups and homogeneous spaces much further, especially in the case of coefficients in a finite field. (For more details on the history of spectral sequences see [McCleary991.) The main theorems of [S erre51] and of this chapter are the following: Theorem 5.1 (the homology Leray-Serre spectral sequence). Let G be an abelian group. Suppose F E-43 is a fibration, where B is path-connected,

5. The Leray-Serre Spectral Sequence 1

135

and F connected. Then there is a first quadrant spectral sequence, {E,r,,,„ converging to II (E; G), with Ep2

Hp (B;h q (F;G)),

the homology of the space B with local coeffi cients in the homology of the fibre of 7r. Furthermore, this spectral sequence is natural with respect to fibrepreserving maps of fibrations.

Theorem 5.2 (the cohomology Leray-Serre spectral sequence). Let R be a E- 43 is a fibraiion, where B is commutative ring with unit. Suppose F path-connected and F is connected. Then there is a first quadrant spectral sequence of algebras, {E*, dr.}, converging to H* (E; R) as an algebra, with

the cohomology of the space B with local coefficients in the cohomology of the fibre of 7. This spectral sequence is natural with respect to fibre-preserving maps of fibrations. Furthermore, the cup product — on cohomology with local coefficients and the product •2 on E2" are related by u •2 y = ( - 1) P'q U when u C 4' and y E g 'q .

The E9-terms of these spectral sequences are expressed in terms of local coefficient systems which are induced by the fibration over the space B. Such systems of coefficients arise naturally in the study of obstruction theory and in the study of sheaves. ISteenrod44] has given the definitive treatment on which we base much of our discussion. In this chapter we derive the homology version of the spectral sequence (Theorem 5.1). It is possible to extend the statement of the main theorems to the case of fibrations of pairs and we will sometimes use the more general statement. The reader is invited to derive the relative cases in the exercises. In order to go quickly to the applications, we postpone the details of the proof to appendices (§5.3). In §5.1, the motivating ideas for the construction of the spectral sequence are given and the E'-term and the differential are identified. We also describe the cohomology version of the spectral sequence and record its relevant multiplicative structure. In §5.2, with machine in hand, we turn to applications. The algebraic examples from Chapter 1 can be applied to yield significant topological results. The classical theorems of Leray and Hirsch and of Borel and Serre are derived in short order with these tools. After such immediate applications, we turn to less general results and compute the cohomology of various Lie groups and homogeneous spaces. We also explore some of the implications in homotopy theory of computations for the based loop space made possible by the spectral sequence. Perhaps the most spectacular results are the generalization of a

136

5. The Leray-Serre Spectral Sequence I

theorem of Marston Morse on the existence of infinitely many distinct geodesics joining two points in a Riemannian manifold and the proof that 7i (S271-1 ) is finite for i > 2n — 1. For the most part, the fibrations considered will have simple systems of local coefficients (that is, one can take them to be constant) and so experience with local coefficient systems is unnecessary (see Chapter 8bis for more subtle results). In §5.3 the reader can find the details of a proof of the homology version of the Leray-Serre spectral sequence and the proof of the multiplicative properties of the cohomology spectral sequence. This requires developing the notion of homology and cohomology with coefficients in a bundle of abelian groups and then identifying H(E 1 ,d') in terms of these notions. The exposition in §5.3 and its subsections owes much to the paper of [Brown, E94 1 . In the rest of the chapter, we have followed [Serre511 and [Bore153]. On first reading, the novice is encouraged to skip §5.3 and go on to further applications of the spectral sequence in Chapter 6. However, §5.1 may disguise the fact that it is the topological structure of a fibration that allows us to identify the E2 -term of the spectral sequence as something familiar. The appendices show that nontrivial results are needed in order to support the powerful machine we exercise in the applications. 5.1

Construction of the spectral sequence

We begin with a fibration: F E —> B. Recall that a fibration enjoys the homotopy lifting property (§4.3), that is, if K is a space and we have a mapping g: K —> E and a homotopy H: K x B so that o g(x) = H(x, 0), then the mapping H can be lifted to a homotopy :KxI—>E, so that itoH = H. K (—,I3)

g >E n7 7'

ir

KxI

B

We take B to be in a "convenient category" of topological spaces, that is, B has the homotopy-type of a CW-complex. We replace B with an equivalent CW-complex in what follows. A CW-complex is equipped with a filtration by skeleta. We lift the filtration of B to a filtration on E by letting J8 = 7 -1 (B( 8 )), the subspace of E that lies over the s-skeleton of B: E

D • • • 9 J*9

B

D ••• D

9

I B s)

J 8-1

I D

9

•••

9

D

• • •

D

Jo 90 13 (°)

D

0.

We use this filtration to obtain a spectral sequence. Putting the methods of Chapter 2 into action, we first obtain an exact couple giving the spectral

5.1. Construction of the spectral sequence

137

sequence. Consider, for each s, the long exact sequence of homology groups for the pair (Js, J8-1 ) and assemble these sequences into an exact couple: H,(Js-1 ;

H,(Js;G)

a

Thus

Epl,q = Hp±q(JP

H,(Js J8-1 ; G)

JP-1 ;G) and the first differential cl1 is given by

a Hr ( js , J 8-1 ; G )

("9-1 ; G)

i(f-1

js-2 ; G),

which is the boundary homomorphism in the long exact sequence of homology /9-1 , js-2) . groups for the triple Alternatively, we can begin with C. (E; G), the singular chains on E with coefficients in G, and filter it by Fs C* (E; G) = im(C* (Js ; G) —> C* (E; G)).

This is an increasing filtration and, by an easy argument (using the compactness and dimension of simplices), this filtration is exhaustive. The inclusions are monomorphisms, so that Epl , = Hp+q(FpC.(E; G) I Fp _iC* (E; G)) = Hp+q(JP ; JP-1 ; G).

The argument of Proposition 2.11 carries over and so we have that the two spectral sequences presented are the same. Since the filtration on E ends over the 0-skeleton of B, this is a first-quadrant spectral sequence. Adding the convergence of the spectral sequence we can summarize the discussion as follows: Proposition 5.3. Given a fibration F E-8 .13 with base space a CW-complex, there is a first-quadrant spectral sequence, {E' 5 , d"}, with HT.±8 ( 7 -1(B ( 8 )) ,

7 -1 ( B (8-1)) ; G)

and d1 = L, the boundary homomorphism in the exact sequence in homology for the triple (7 -1 (B (8) ), 7T -1 (B (s-1) ), 7 -1 (B (8-') )). The spectral sequence converges to H8 (E; G).

For cohomology with coefficients in a commutative ring R, we have the analogous long exact sequences for the pairs (J8, J') and so the exact couple t*

Hr (Js; R)

(js-1;

Hr+1 (Js; Hr+

( „IS

J8 _i ; R)

138

5. The Leray-Serre Spectral Sequence I

and a spectral sequence with d1 = A, the boundary homomorphism in the long exact sequence in cohomology for the triple (Js, Js– ', J8-2 ). The dual filtration of C*(E; R) is given by Fs C* (E; R) = ker(C* (E; R)

C* (Js-1 ; R)),

and the dual version of Proposition 5.3 holds for cohomology. The next step in establishing Theorems 5.1 and 5.2 is the determination of the E2 -term. Consider the simplest case of a trivial libration, E = BxF. By induction over the skeleta of B, one can show that (Js Js–i) _ (B(8) , B(s-1)) >< F in this case and so the Ktinneth theorem can be applied to show that El19

,9

Cell(B) 0 Hq (F; G)

with given by cr = acen 0 1. Thus 4,1 Hp (B; Hq (F; G)). In the case of an arbitrary fibration, the 'twisting' of the fibre and base spaces in the total space is a global and possibly nontrivial phenomenon and so this prevents a simple expression for El- and hence for E2 . In §5.3 we define the chains on a space with coefficients in a bundle of groups. The relevant bundle of (graded) groups associated to a fibration is given by 1-15 (F; G) = {H.(7-1 (b); G) b E /31,

equipped with the collection of isomorphisms, {h[)■] H.(7 -1 (b2); G) —>

Hs (7 -1 (bi);

AE

(B,

b2)}

indexed over the homotopy classes of paths, A, in B from b 1 to b2 . The main results of §5.3 are that Epl , q Cellp (B; H q (F; G))

and that = 0"11 on these groups. This establishes Theorem 5.1. In the examples of §5.2 a particular class of bundles of groups often occurs for which all of the isomorphisms h [A] may be taken to be the identity. This is called a simple system of local coefficients for which E13l ,g

Cellp (B; Hq (F; G)), cl, =0"11 ,

that is, the contribution of the varying fibres gives the system of constant groups Hq (F; G). If B is simply-connected, then any fibration over B leads to a simple system of local coefficients (Proposition 5.20); thus we can identify the El- - and E2 -terms of this spectral sequence directly for a large class of examples. We state Theorem 5.1 for this case as follows:

5.1.

Construction of the spectral sequence

139

Theorem 5.4. Let G be an abelian group. Given a fibration, F where B is path-connected, F is connected, and the system of local coeffi cients on B determined by the fibre is simple. Then there is a spectral sequence, with Ep2, Hp (B; Hq (F; G)), and converging to 11.(E; G).

Multiplicative properties of the spectral sequence

Theorem 5.2 asserts that, for a commutative ring R and a fibration F E B with B path-connected and F connected, there is a spectral sequence of algebras with L' HP (B; 7 1.' (F; R))

and converging to H* (E; R) as an algebra. First a word about the proof: For the tech or Alexander-Spanier cohomology theories, the multiplicative structure is carried along transparently in the construction of the spectral sequence and so we get a spectral sequence of algebras directly with convergence to H* (E; R) as an algebra aLeray501, [Bore1511). The result for singular theory, however, is more difficult—it is one of the technical triumphs of Serre's celebrated thesis ([Serre51]). We present a proof of the product structure for this spectral sequence in §5.3 following the exposition of [Brown, E94 1 . The interested reader should consult the thesis of [Serre51], not only for the first proofs of Theorems 5.1 and 5.2, but also for a model of exposition. Alternative presentations appear in Chapter 6, in exposé 9 of the 1950 Cartan Séminaire by [Eilenberg50], and in the books of [Hu591, [Hilton-Wylie601, and [Whitehead, 0W781. An exposition of the result for 'Cech theory appears in the book of [Bott-Tu82]. By introducing some simplifying hypotheses, the spectral sequence takes a manageable form. Proposition 5.5. Suppose that the system of local coefficients on B determined by the fibre is simple, that F is connected, and that F and B are of finite type; then, for afield k, we have HP (B; k) ®k H' (F; k).

We leave the proof to the reader with the hint that you should apply the Universal Coefficient theorem. Suppose that the system of local coefficients is simple and that B and F are connected. Then HP (B; H° (F; R))

HP (B; R)

EL;'q H° (B; Hq (F; R)) Hq (F; R). and For any bigraded algebra, like both E2' and E2' 4' are subalgebras. ''()

0

140

5. The Leray-Serre Spectral Sequence I

Proposition 5.6. When restricted to the subalgebras E2*.0 and E20 ' *, the product structure in the spectral sequence on E2" coincides with the cup product structure on H* (B; R) and H* (F; R), respectively. Furthermore, if for all p, a, HP(B; R) and Hq(F; R) are free R-modules of finite type, and the system of local coeffi cients on B is simple, then E24" H* (B; R) ®R H* (F; R) as a bigraded algebra.

The proof of the first part of this proposition is postponed to Example 5.E where it follows from the naturality of the spectral sequence. The rest of the proposition is an application of the Universal Coefficient theorem. As the reader begins the applications in §5.2, we remind her or him of the examples in Chapter 1; the techniques sketched there can be applied to the Leray-Serre spectral sequence with considerable success. 5.2 Innnediate Applications In this initial collection of applications we interpret topologically some of the formal consequences of having a spectral sequence. The second set of applications will be concerned with the problem of computing the cohomology of certain Lie groups by using the Leray-Serre spectral sequence as applied to the fibrations that result from quotients by subgroups. Finally, we investigate the path-loop fibration over a space. We begin with some dimensional arguments. Example 5.A. Suppose F E-43 is a fibration with B path-connected and the system of local coefficients on B induced by the fibre is simple. If R is Noetherian and two of the spaces F, E, or B have cohomology a finitely generated R-module in each dimension, then the other space also has cohomology finitely generated in each dimension.

CASE 1: Suppose B and F have cohomology that is finitely generated in each dimension. The module Hn(E; R) is associated to Ep:q via the P±q =n associated graded module to the filtration on H*(E; R) and so it suffices to show that each E is finitely generated over R. We begin with an E2 -term given by 4' H (B; 11q (F; R)) which is finitely generated by an argument using the Universal Coefficient theorem. Since Er is a subquotient of 4, it is finitely generated because R is Noetherian. Similarly EV, EE'q, , and so finally Elc q are all finitely generated. CASE 2: Suppose E and B have cohomology that is finitely generated in each dimension. We show by induction that Hri(F; R) is also finitely generated. Since H° (F; R) = EY = H° (E; R), it i s finitely generated. Let n be the least integer such that Hri(F; R) is not finitely generated. Then E2° 'n = Hn(F; R) is not finitely generated and so neither is E220,1 — H 2 (B; (F; R)) E(3) 'n = ker d2 : E(2)'n

,ED

5.2. Immediate Applications

141

because the target module is finitely generated by the induction hypothesis: E33 '' -2 is a subquotient of H 3 (B ; Hn-2 (F; R)). so it is finitely generated and hence .F°' = ker cI3 E' E33 'n-2 cannot be finitely generated. Continuing tti is arguinent, we get that .E(2f is not finitely generated but this is a contradiction, since .Egf is a subquotient of H"(E; 11). CASE 3: Suppose E and F have cohomology that is finitely generated in each dimension. By an argument similar to the one in case 2, we derive a contradiction if, for some n, H(13; R) = E;'° is not finitely generated. CI

Suppose the assumptions of Example 5.A hold and that R = k, a field. Then we have 41 HP (B; k)o k Hq (F; k) and we can speak of the Poincaré series and the Euler-Poincaré characteristic. We refer the reader to Example 1.F where we show that P(E t) P (B t) P (F, t) and x (E) = x(B) • x(F). whenever these expressions are meaningful. Example 5.B. Suppose F EL43 is a fibration with F connected, B of finite type, path-connected, and the system of local coefficients on B induced by the fibre is simple. Suppose k is a field, that Hi (B; k) = {01 for i > p and Hj (F; k) = {0} for j > q. Then H'(E; k) = {O} for i > p + q and HP+q(E; k) = HP (B; k) 11 7 (F; k). These assumptions imply that g2.'s fir (B;k)Ok Hs (F;k) and furthermore, that E," is nonzero only in the box pictured.

Since the spectral sequence converges to 1-1* (E; k) and, k is a field. Hn(E; k) `L' er+8,E,-: as a vector space implies that Hi (E; k) = {O} for i > p q. Also from the picture, no differential can affect ElPg , so it persists to El,",;q HP (137 k) Hq (F; k). Since this is the only vector space contributing to HP' (E; k), the result follows. A corollary to this formal consequence of the data is the following seminal result of [Borel-Serre50].

142

5. The Leray-Serre Spectral Sequence I

Theorem 5.7. Suppose F —>B is a locally trivial fibration with B a polyhedron and F connected. Then B and F are acyclic spaces (that is, fl* (B) {0}

TI* (F)).

PROOF: By the tail end of the homotopy exact sequence, • • • —> 7ri (Rn ) —> 71(B) —>

7ro(F) —> • • •

we can see immediately that 7ri (B) = {0} and so the system of local coefficients induced by B on F is simple. Observe that Hi (B) and Hi (F) are trivial whenever i > n. To see this we make some simple observations from dimension theory: (1) F is a subset of V and (2) B has a system of neighborhoods, {U}, so that 7 -1 (U) is homeomorphic to U x F for each U in the system.

Since 7r -1 (U) is a subset of V, U must have dimension < n. The argument of Example 5.B applies to show that HP±q (Rn )

HP (B) Hq (F)

Tor(HP+ 1 (B), H4 (F)))

HP (B) Hq (F)

when p and q are the greatest nonzero dimensions in which B and F, respectively, have nontrivial integral homology. Since Rn is acyclic, p + q = 0 and we conclude that B and F are acyclic. (For the interested reader, we note that the argument of [Borel-Serre501 is given in terms of Alexander-Spanier cohomology with compact supports and original spectral sequence of 1Leray501.) Another example of this kind of argument, using the spectral sequence as guide, is given by sphere bundles over spheres with spheres as total space. That Sc. In this case, using cohomology with is, we have a fibration Sm Sn coefficients in a field k, our E2-term has the following form:

143

5.2. Immediate Applications

There is a nontrivial differential since the vector-spaces

4,- = .11° (S t ; Hrn (S";k))

(Stn ; k) = k

and Ef,'0 (5'; H° (5"; k)) = k must be mapped isomorphically to each other if we are to have the cohomology of a sphere at E. Since differentials have bidegree (r, 1 — r), we conclude that m = e 1 and, following Example 5.E, n = f + m = 2e —1. For = 2,4 and 8, the Hopf fibrations are examples of such fibrations. —

ELB is a libralion Example 5.0 (the (;ysin sequence). Suppose F with B path-connected and the ,5ystent of local coefficients on B induced by the fibre is simple. Suppose further that F is a homology n-sphere, that is, H(F) .14(5), for some n > 1. Then there is an exact sequence: Hk (B; R) where 7(u) = z R, then 2z = O.

Ern+k+1(B ; R )

B-1(E;

.R)

B k+1 (B; R) —)•

for some in Hn+1 (B; R) and, i f n is even and 2 0 in

R when m = 0 or n, Since F is a homology n-sphere, Hm(F; R) and H(F; R) is trivial in the other dimensions. With B path-connected, the .E9-term of the spectral sequence can be pictured:

By the placement of holes, E2 E3 • • • En+1 and H(En-Fi., 4+1) En+2 E. By the argument of Example 1,D there is an exact sequence Ek 2

k+n+1 ,0 2

arark-kn-o. (E; R)

an-F1

k +n ,0

Suppose h is a generator of frn (F; R). Since we have a simple system of local coefficients and F is a homology n-sphere, we can write E:441 '-=`'

(B; R) 01) ED (.1-1' (.8; R) h).

144

5. The Leray-Serre Spectral Sequence I

Let z be the element in Hn+1 (B; R) such that 4+1(1 0 h) = z 0 1. (Notice that c/n±i (x o 1) = 0 for dimensional reasons.) Proposition 5.6 tells us what kind of spectral sequence of algebras we have, and so 1rdegsdn±i(x

h) = dn+1((1

= dn+1(1 = (z 01)

(x 0 1)) dr,+1(x 0 1)

(x 01) + (-1) n (1 (x 01) = (z x)

Identifying E2" = Hk(B; R) O h and E2n±k±l '0 = Hn+k+1(B; R) 1, we 7(x) = dn±i(x 0 h) and we get the exact sequence as described. defin If n is even, then we use the Leibniz rule and the fact that h ---- h = 0 to compute: O = dn+ ,(1 0 (h --- h)) = cln+1 ((1 = (z 0 1)

(1 g h) + (-1)n(1

--- (1 0 h)) h)

(z 1)

= 2z e h.

Since 2h 0, we must have 2z = O. The long exact sequence in this example is called the Gysin sequence (IGysin411). With this sequence, we can compute H* (CP(n); R) as an algebra. The n-dimensional complex projective space, CP (n), sits in a fibration with spherical fibre, Si- 52 n+ 1 CP(n). We have CP (1) S2 and CP(n) is simply-connected for all n. Thus the system of local coefficients is always simple. We next show that (p( n) ; R )

R[x2]/ (z21+1 ) ,

the truncated polynomial algebra of height n+ 1 on one generator of degree 2. First notice that, x2 = -y(1) generates H2 (CP(n); R) from the initial part of the Gysin sequence: Ho ( cp(n) ; R ) 1, H 2 (cp(n);

H2 (S2

';

Moving on a little further, we find, for n> 1, the short exact sequence {0} —> H 1 (C P (n); R) 1> H3 (C P (n); R)

{

0}

n+ R) for k = 2, 3. Since CP (n) is where the trivial modules are H' (921; simply-connected, Hi- (CP(n); R) is trivial and hence, so is H3 (CP(n); R). By induction, suppose that (x2) k generates H2k (C P (n); R). Then consider the portion of the Gysin sequence,

R 2k (c p (n); R ) L R 2k+2 (c p(n) ; R)

R 2k+2 (s2n+i R)

5.2. Immediate Applications

145

( x2 )k+1 When k < n, we have that is an isomorphism, so x2 -- ( x2 ) generates If 2k +2 (CP(n); R). In odd dimensions, the pattern of trivial modules continues. Finally, when k = n, we have the short exact sequence. 0

H2n+1 ( ( P (n); R)

H2' 1 (S21; R) 4 H an (cp(ro; R) 11 2,,+2(cp(n); R)

O.

Since .12P(n) is a 2n-dimensional manifold, 7 is the trivial homomorphism and Q is an isomorphism. Since -y((x9)") = 0 and /((x2 )') = 4 +1 . we conclude (x2) n+1

O.

We leave it for the reader to derive the Wang sequence ([Wang49]) for a fibration F E B with B, a simply-connected homology n-sphere and F path-connected: • • • Hk (E; R) Hk (F; R)—) ° H"±1 (F; R) Hk +1 (E; R) • • • In this case, if n is even, the mapping 0 can be shown to be a derivation and, if n is odd, an antiderivation (that is. 0(x — y) = 0(x) — y + x (I(y)).

Example 5.D. Suppose F E L B is afibration with B path-connected and the system of local coefficients on B induced by the fibre is simple. Suppose further that H' (B; R) = tOl for 0 < i

146

5. The Leray-Serre Spectral Sequence I

In total degrees less than p + q the only possible nonzero differentials are of the form : Hi-1 (F; R) = EV -1 —> 4 ° = Hi (B; R).

Arguing as in the derivation of the Gysin sequence, we get a diagram of short exact sequences spliced together:

NI

Hi-1 (E; I?)

0

iN 0

0 H' (E; R)

0

0

2

The sequence remains valid until j = p + q. Making the identifications E2° '•1— ' H1-1 (F; R) and BAB; R) gives the desired sequence. As we will see in Example 5.E, the homomorphisms in the sequence are given by the differentials, di, plus 7F * and i* from the fibration. Furthermore, in §6.2, we will identify the di with the geometric homomorphism called the transgression. Without the connectivity assumptions, the resulting short exact sequence has five terms and is given in Example 1.A. The reader can establish the dual sequence in homology where the analogy with the long exact sequence in homotopy is more apparent. This sequence has become known as the Serre exact sequence ([Serre51, Chapitre III, §41). We record some immediate corollaries. Corollary 5.8. For a fibration F E L B with B path-connected and for which the local coefficient system is simple, (a) if E is acyclic and Hi (B; R) = {0} for 0 < i < p, then Hi (F; R) Hi+1 (B; R) for i <2p — 2; (b) if B or F is acyclic then i: F —> E or 7T: E B, respectively, is a weak homotopy equivalence.

We return to these ideas later in this section where they apply to the path-loop fibration.

Example 5.E, in which we exploit naturality.

Consider the descending filtration on Hq = Hq (E; R), Hq = F° Hq D F1 Hq D • • • Fq-1 Hq D FHq D F 4+1 Hg = {0},

5.2.

147

Immediate Applications

n Hq I Fn+1 H Since Fg+1- I-Ig and its associated graded module, E — n = {0 } , we have Ee C Hg(E; R). Furthermore, there is the short exact

sequence, 0

Fl Hq

Hq(E; R)

E

0.

The modules along the bottom row of the E2 -term of a first quadrant spectral sequence are determined by quotients 40 by the image of the incoming differential. Thus we have a series of quotients (here » denotes an epimorphism): Eq

)

> wq,0

wq,0

' 13

The terms of the modules along the "y-axis" of the E2-term of a first-quadrant spectral sequence are the kernels of successive differentials, and so we get a sequence of inclusions: .E((q

q+1

Oq

C

C • • • C E3 '

Oq C E2 ' .

These facts are formal consequences of having a first-quadrant spectral sequence. ir

Theorem 5.9. Given a fibration F E B with B path-connected, F connected, and for which the system of local coeffi cients on B is simple; then the composites Hq(B; R) = ET °

> ET °

)>

» EQ °

and Hq (E; I?)

» Eqq ), =

»E = E(1°Z c E q° , q c • • • c

c Hq (E; R) = Hq(F; I?) are the

homomorphisms 7r* : H' (B; R)

H" (E; R)

and

: H' (E; R)

Hg (F; R).

PROOF: Consider the diagram of fibrations:

B

B

By the naturality of the Leray-Serre spectral sequence, we get induced mappings of the spectral sequences Er (B , B , *)

ir*

Er (B , E , F)

Er (*, F, F),

148

5. The Leray-Serre Spectral Sequence I

which converge to i* and 7F * at E. However, E2 F, F) consists of the column E20 ' * = H* (F; R) and so collapses at the E2 -term. Also E2 (B , B, is the single row, E2*'0 = H* (B; R) and the induced mappings of spectral sequences project H° (B; H* (F: R)) onto H* (F; R) and inject H* (B; R) into H* (B; H° (F; R)). In order to prove the theorem, it remains to consider how the mappings i* and 7F* are approached by the successive homologies in the spectral sequence and this is given in terms of the inclusions and quotients in the theorem. Some immediate consequences of the theorem have already been mentioned. Since the induced maps are morphisms of algebras, we have established Proposition 5.6 on the product structure of E2Q * and E*2 '° • Furthermore, the corresponding mapping in the Gysin sequence is given by 7F* and in the Serre exact sequence by 7F * and i*. A topological invariant of the inclusion of F in E is the homomorphism i*. We say that F is totally nonhomologous to zero in E with respect to the ring R if the homomorphism i* : H* (E; R) —> H* (F; R) is onto (introduced by [Samelson411). In the case of a trivial fibration, this holds for obvious reasons. We can deduce something of a converse. Theorem 5.10 (the Leray-Hirsch theorem). Given a fibration F E B with F connected, B of finite type, path-connected, and for which the system of local coefficients on B is simple; then, i f F is totally nonhomologous to zero in E with respect to afield k, we have H* (E; k) H* (B;k)

Ok

H* (F; k)

as vector spaces.

PROOF: The theorem follows immediately from the following stronger assertion. Fact. F is totally nonhomologous to zero in E with respect to k if and only if the spectral sequence, Er (B , E, F), collapses at the E2 -term. To establish this fact, examine the expression for i* given by Theorem 5.9: i* : Hq (E; R)

=

Eq"

••• = Hq (F; R).

If the spectral sequence collapses at the E2 -term, then all of these inclusions are equalities and i* is onto. Conversely, if i* is onto, then all of these inclusions must be equalities and so dr, restricted to the y-axis, must be zero. At E2 however, 4' = 4'0 E 'q since k is a field and d2 on 4'0 is already zero. Since d2 is a derivation with respect to this representation, d2 = 0 and E3 = E2.

5.2. Immediate Applications

149

The same argument applies to d3 and, continuing in this fashion, we establish that the spectral sequence collapses at E2. By Theorems 5.9 and 5.10 it follows that if i* is onto, then 7r* is injective. It was conjectured that the converse also held. We sketch a counterexample due to G. Hirsch (see IBore1541) showing that 77* can be injective without i* onto. A systematic set of counterexamples was found by [Gottlieb771. Hirsch's example begins with the Hopf fibration : S 7 —> S4 (see §4.3), which has fibre S3 • As mentioned in Example 5.B, the Leray-Serre spectral sequence for y has a nonzero differential, d4 , taking the generator of h-- * (S3 ; R) to the generator of ii* (84 ; II). Consider the mapping of degree 1 from S2 x S2 > S4 that can be constructed as the composite S 2 x s2 s2 A s2 s4 . Pr Finally consider the projection S2 x S' S' onto the first factor. Put the ingredients together in the diagram: —

s3

53

E

S7

t

iv

P

S2 x

S2

deg

1

s4

Prl S2

where E is the total space of the fibration gotten by pulling y back over the degree 1 map. By naturality, the spectral sequence for (S 3 , E, S 2 x S 2 ) does pri op E —> S2 . Let not collapse; d4 O. Consider the fibration S3 x S2 s s2 s2 x S2 split pri . Examine s* on the generators of the E2-term of the spectral sequence for (S3 x S2 , E, 32 ) . For dimensional reasons, this splitting implies that (pr 0 pr is injective. However, the spectral sequence does not collapse since we know already that H* (E; R) is not isomorphic to H* (S 2 x S 2 x S 3 ; R). We leave it to the reader to determine the nonzero differential.

Particularfibrations The importance of Lie groups in homotopy theory cannot be overemphasized. It is often necessary to compute the homotopy-theoretic invariants of various Lie groups and homogeneous spaces. Their structure naturally leads to some useful fibrations where the Leray-Serre spectral sequence can be applied. It would be impossible to sketch all of the applications in topology where Lie groups and homogeneous spaces play a crucial role. For a sampling of some of these results we refer the reader to the accounts of [Steenrod511, [Bore1551, [Milnor-Stasheff741, [Mimura-Toda911, and [Dwyer-Wilkerson941. In the next

150

5. The Leray-Serre Spectral Sequence I

few examples, we compute the cohomology algebras of some of these spaces as much to illustrate some further techniques with spectral sequences as to obtain these important data. These computations are based on the thesis of [Bore1531. Example 5.F, in which we compute H* (SU(n); R).

The classical Lie group SU(n) is the group of (n x n) complex-valued unitary (AA' = I) matrices of determinant 1. If we fix a vector Yo G C n , then we define p: SU(n) s2n-i by sending a matrix A to Ah. This mapping can be proved to be a fibration (see §4.3) with fibre the subgroup of SU(n) that fixes Yo, which is SU(n - 1). The group SU(2) may be identified as the unit sphere in R4 or S3 (by quaternionic representation). We next prove that H* (SU(n); R) A(x3 , x5, . , x2_1), the exterior algebra on generators, xi, where deg xi = j. We proceed by induction on n. For n = 2, H* (SU (2); R) H* (S 3 ;

= A(x 3 ). ,

Consider the fibration SU(n - 1) <- SU(n)s 27-1-1 and apply the Leray n- R) as an exterior Serre spectral sequence. Suppose Y2n-1 generates H* (s21; algebra. By induction, we have II* (S U( 71

—

1); R)

A(X3 1 X51 • • • X2n-3)•

Since this is a spectral sequence of algebras, we need consider only the algebra generators in describing differentials. For n > 2, s2n -1 is simply-connected and so the system of local coefficients on the base space is simple. Furthermore, by an application of the Universal Coefficient theorem, we have E2"

H* (B; R) 0 H* (F;

A(y2„_ i ) ®

A (x3,

. . . , x2n-3).

The algebra generators are found in bidegrees so that all of the differentials are zero and the spectral sequence collapses at the E2-term. We argue as in Example 1.K to conclude that II* (SU(n); R) '="'

Tot E*

A(X3, X5, • • • W2n-31 Y2n-1)

and the induction is completed. We can extend the computation to H* (U(n); R), where U (n) is the group of linear transformations of Cn which preserve the complex inner product. The group U(n) relates to SU(n) via the fibration: inc

det

SU(n) —> U(n) —>

5.2. Immediate Applications

151

and furthermore, U(n) U(1) x SU(n)

x SU(n). Thus, H* (U(n); R) the symplectic group, Sp(n), of linear transformations of Ein (quaternionic n-space) that preserve the inner product. It is left as an exercise to show that H*(Sp(n); R) A(x 1 , x3, • • • , x2,1). These methods also apply to

A(x3,x7, • • • , x4,1)•

A space that is important in the study of cobordism (see Chapter 9) is the infinite special unitary group, SU, which is the direct limit (union) of the

sequence of inclusions

SU(2) c SU(3) c • • • c SU(n — 1) c SU(n) c • • • c SU. By choosing a unit vector Yo in S 2n.— ', we can realize the usual inclusion of s2n —1 SU(n —1) c SU(n) as the inclusion of the fibre SU(n-1) SU(n) By Theorem 5.9, we can describe i* H* (SU (n);

—> H* (SU (n — 1); R)

as the mapping taking xi to xi except on x2n_1 which is sent to zero. Since each i* is an epimorphism, lim l H* (SU(n); R) = {0} (§3.2) and so H* (SU; R) A({x2i_i i = 2, 3, ... 1). Similar results hold for the analogous groups U and Sp. We take up the real case of SO(n) in Example 5.H. Example 5.G, in which we compute H* (Vk

(C n ); R).

The Stiefel manifolds Vic (11n ), Vk (Zn ), and Vic (Hn ) consist of the orthonormal k-frames in each n-dimensional space over the given (skew) field. These spaces play an important role in the study of characteristic classes (see [Milnor- Stasheff741 and §6.2) and in the study of vector fields on manifolds; they are among the simplest examples of homogeneous spaces. In this example, we compute H* (Vk(U n ); R) using the Leray-Serre spectral sequence (see §7.2 for another approach). We will take up H* (Vk(R n ); R) in Example 5.H, and we leave the computation of H* (Vk (E in ); R) to the reader. Let e, Cn denote the i th elementary vector, ei = (0,

, 0, 1, 0,

, 0),

with the 1 in the ith place. Consider the mapping (for n> 2), SU(n) Vk (Cn ) given by applying the matrix A to each ei in turn to get the orthonormal k-frame (Ae l , Ae2 , , Ae k ). This map is clearly onto and continuous and each given k-frame can be varied by the action of A on the n — k remaining vectors in the standard n-frame for Cn . Therefore, the fibre over a point in Vk (C n ) is a copy of SU(n — k). By including a matrix in SU(n — k) into SU(n) as the bottom right hand (n — k) x (n — k) block of an otherwise n x n identity matrix, we

152

5. The Leray-Serre Spectral Sequence I

see that Vk (Ca ) is homeomorphic to SU(n)/SU(n - k). Thus we have the fibration SU(n - k) —> inc SU(7l)

(Cn )

It is immediate from the definitions that V1 (Ca ) = S2n - ' and that Vn (Cu ) = SU(n). With the identification of Vk (C a ) as a homogeneous space. we can apply a little group theory to obtain some short exact sequences that give us new fibrations. The inclusion of subgroups SU(n - k - 1) c SU(n - k) provides us with the fibration 0 SU(n - k)/SU(n - k - 1) -> SU(n)/SU(n - k - 1) -> SU(n)/SU(n - k) ine V,1 which may be identified as 82(n—k)-1 —>

Proposition 5.11. H* (17k (C n ); R)

A(x 2( n -

(C)

0,

V(C) .

+11 X2(n—k)+35 • • • X2n-1)•

PROOF: We proceed by induction on k. For k = 1, H* (VI (Ca); R) = B -* (s2n-i )= A (x2n _ i ) and the proposition holds. Now suppose the statement holds for the value k. The Leray-Serre spectral sequence for the fibration (s2(n-k)Vk+ i(Ca ), V(C)) has E2-term given by E2*,*

(vk (cn) ;

( s2(n-k)-1 ; R))

A(x2(n-k)+1, x2(7,0+3, • • . , x2r i_1) 0 A(y 2 (n _o_ i )

by the Universal Coefficient theorem and the fact that the system of local coefficients is simple. We obtain the following diagram of the algebra generators in the E2-term of the spectral sequence:

Y2(n k) 1 -

1

-

x 2(n-k)+1

x 2(n-k )+3

x 2n-1

For dimensional reasons, the spectral sequence collapses and so A(Y2(n—k)-11X2(n—k)±11 • • • X2n-1)•

Once again, following Example 1.K, H" (Vk+1(C n ); R) this completes the induction.

e

Eg,(1 and

P+q=n

5.2. Immediate Applications

153

Example 5.H. We compute H* (17k (IRn); 72) and, by letting k = n, we compute H*(S0(n); F2).

The new feature about these computations is how naturality can determine certain differentials. We proceed a step at a time. STEP 1: Compute H* (V2 (Rn)). We start with the libration, S 2 V2 (Rn ) —> (Rn ) that leads to an E2-term in the Leray-Serre spectral sequence as in the diagram

n 2Z -

n-1

o

The only possible nontrivial differential may be written dn_ i : H

2 (F) — H' (B).

Consider the tangent bundle over Sn-1 , Rn-1 TS n-1 Sn — ', and the associated sphere bundle of unit tangent vectors, Sn -2 T0 S-' sn-1. By the geometry of the sphere elements in T0Sn -1 are pairs of vectors in Rn , (u, v). with u in Sri' and y orthogonal to u, that is, (u, y) is a 2-frame in Rn The mapping ToSn —i is simply projection on the first factor so we get an isomorphism of fibrations sn- 2

Sn-2

V2 (Rn )

ToSn 1

1 (Rn )

=

Sn -1 .

Thus to study the spectral sequence for V2 (Rn ), we replace it with the spectral sequence for Sn-2 T0 S' sn-1. As a sphere bundle, we can apply the Gysin sequence for which the relevant portion is given by HO (ST1-1)

TITL-1

(Sit —1)

1_171-1

(To ST1-1)

1_11 (ST1-1)

154

5. The Leray-Serre Spectral Sequence I

The unit 1 G H° (S') goes to the cohomology class z following Example 5.C. If n is even, then the class z satisfies 2z = O. Since Hn -1 (Sn -1 ) Z, this is only possible if z = O. When n is odd, we need a little more information about the class z. In fact, for a vector bundle like the tangent bundle, z is identified with the Euler class of the bundle. (See the accounts of [Milnor-Stasheff741 and 11-lusemoller661.) Since the class z is a multiple of the generator of H' (Sa — ') and the cap product of z with the canonical generator of Hii_ i (S71— ') gives the Euler characteristic for Sn', we see that z is twice the generator and dn _i is multiplication by two. It follows (for n > 4):

H* (V2 (Rn )) =

Z., for * — 0, 2n — 3, if n is odd { Z/27_7„ for * = n — 1, 0, elsewhere. if n is even

Z, I. 0,

for * = 0, n — 2, n — 1, 2n — 3, elsewhere.

As a corollary, from the Universal Coefficient theorem, we have (for n > 4) H* (V2 STEP

=

{ 72,

0,

for * = 0, n — 2, n — 1, 2n — 3 , elsewhere.

2: Some comments on simple systems of generators.

Definition 5.12. Suppose H* is a graded commutative algebra over R. A set of elements {xi i = 2. 3, ... } is called a simple system of generators if the elements 1 and xi, x i2 x i, with j 1 < i2 < •
The notion of a simple system of generators was introduced by [Bore153]. 1f H* is an R-algebra with a simple set of generators, then the homogeneous degree m part of H* is a free R-module on the collection of monomials xi, • x 2 • • • xi, with j1
155

5.2. Immediate Applications

STEP 3: H* (S 0(n); 72) has a simple system of generators wheredegx=i.

{x1,x2, • • •

We proceed by induction. For n = 2, S 0 (2) = S' and the statement holds. inc

Consider the fibration SO (n - 1) SO(n) Sn -1 . The spectral sequence, by the induction hypothesis, has a lacunary placement of algebra generators with dn_ i the only possibly nontrivial differential. To show that dri_ i is the zero differential we consider the morphism of fibrations SO(n - 1) SO (n) Sn

Sn-1 .

V2(Rn )

This morphism is a consequence of the following diagram of short exact sequences of groups: 1

1

I

1

1 — SO(7/ - 2)

SO(n - 1)

SO(n - 1)/S0(n - 2)

1

1 — SO(72 - 2)

SO(n)

) SO(n)/S0(n - 2)

1

S0(n)/S0(n - 1)

1

1 SnI

1 1 In the morphism of spectral sequences induced by the morphism of fibralions, the generator of H* (Sn -1 ;7 2) is mapped to yn-1 and the generator of H* (Sn -2; 72 ) to a:n-2. In STEP 1, we established that cin _ i for the spectral sequence associated to (S n-2 , V2 (Rn ), Sn-1 ) is zero modulo 2. By naturality, d_ i is zero in our original spectral sequence. Thus the spectral sequence collapses at the E2 -term. Furthermore, by Theorem 5.13, E2*,* has a simple system of generators, {xi, x2, • , Xn-21Yn-1 } and the induction is completed. STEP 4: H* (Vk ( ln );7 2) has a simple system of generators {Xn—k, Xn—k+11 • • • Xn-111

where deg x i = j.

As in STEP 3, we begin with the libration cm

—k-1

inc-14+1(

Vk(IP

)

156

5. The Leray-Serre Spectral Sequence I

By induction and the spectral sequence associated to the fibration, we get a diagram of algebra generators, with, once again, one possibly nontrivial differential, dri—k •

The assertion in STEP 4 is proved if we can show that dn _k is the zero differential. To settle this, we consider the inclusion Vk_ i(Ill n-1 ) C 14(R 1 ) given by considering each (n — 1)-vector as an n vector with 0 last entry and sending the (k — 1) - frame (vi, v2, , vk_1) to (vi, v2, ... ,vk_i, en ) (where en is the vector (0, , 0, 1)). With this injection we get a sequence of maps of librations: sn-k-1

sn-k-1

V2 (Rn -k+1)

v3 (Rn-k +2)

sn-k

v2 (Rn -k+2

sn-k-1

Vk(R n

1)

Vk-i(R n-1 )

sn- k -1

irk+ 1 (Rn

)

Vk (Rn ).

Each inclusion Vk_ (R n-1 ) C VORn ) is the inclusion of the fibre in the fibration 14_ 1 (Rn-1 ) V(R) —> S n-i and so, by step 3, the class X n _ k is mapped to the corresponding x n _k by i*. By naturality, we have reduced the question of the collapse of the spectral sequences for all k to computing dn _ k in the spectral sequence for the particular fibration (Sn —k ', V2 Pr -k+1 ), Sn-k ) that was shown to be zero modulo 2 in STEP 1. As in STEP 3, the induction is completed with Theorem 5.13. Loops on a space Let X be a topological space. We associate two important spaces to X as follows: Suppose * is a basepoint in X. Let PX = {A: [0, 1] —> X A is continuous and A(0) = *1 denote the space of paths in X based at *. Let = {A : [0,11 —> X A is continuous and A(0) = A(1) = *1, the space of based loops in X at * with the compact-open topology. The evaluation mapping evi : PX —> X given by evi (A) = A(1) has fibre S 2X. In §4.3 we

157

5.2. Immediate Applications

showed that evi : P X —> X has the homotopy lifting property and so we have

a fibration QX Px —> evl X.

The space P X is contractible with the immediate consequence that, for all > 0,

7ri (52X) '=•' r

i (X) (Corollary 4.31).

An important feature of QX is that it is an H-space, that is, there is a multiplication, tt: QX x QX QX, A2)(t) =

* A2(t) = {

if 0 < t <

(2t),

if 7!): < t < 1. Up to homotopy, /1 has an identity and is associative making the cohomology algebra of QX a Hopf algebra. We also recall that 71(QX) is abelian as a consequence of the H-structure. A2 (2t -

1),

Example 5.1. Suppose X is q-connected. Then, H1 (X; R) H' (X; R) for i < 2q— 2. The analogous isomorphism, Hi(S2X; R) Hi +i(X; R) holds in homology for i < 2q — 2.

This follows by the acyclicity of P X and Corollary 5.8, the Serre exact sequence. We discuss these isomorphisms in §6.2. We apply these ideas to prove the classical result. Theorem 5.14 (the Hurewiez theorem). If X is a connected space so that ri (X) = {0} for 1 <j < q, then Hi(X) = {0} for 1 <j < q and 7q (X) H,(X). PROOF: We proceed by induction on the connectedness of X. If X is simplyconnected, that is, 7ri (X) = {0} for i < 2, then 7r2 (X)

7r i (QX)H1 (X) 112(X),

where the middle isomorphism is the classical theorem of Poincaré and the fact that the fundamental group of an H-space is abelian, and the last isomorphism follows from the Serre exact sequence. Suppose the theorem is true for spaces that are (q — 1)-connected and X is q-connected (q > 2). By the isomorphism, 7r_1(X) Tri (X), we have that S2X is (q— *connected and so by induction, Hi (QX) — {0} for 0 <j < q— 1 and 7rq _ i (QX) Hq _ i (QX). Finally we use the isomorphisms implied by the fibration in homology and homotopy 7,(X) 7,_ 1 (S2X) H g _ i (S2X) H,(X).

For 1 < j < q, since i < 2q —2, Hi (X) H i_ 1 (S2X) = {0}, and Hi (X) = {0} because 7ri (X) =

01.

Example 5.A for the case of the path-loop fibration implies that if R is Noetherian and H* (X; R) is finitely generated in each dimension, then so is H* (X ; R).

158

5. The Leray-Serre Spectral Sequence I

Example 5.J, in which we consider a remarkable theorem of [Morse34]. Let P(X,a,b) denote the space {,)1/4 : [0,1] X I MO) = A(1) = b} of paths joining a to bin X. a subspace of map([0,1], X). If we suppose that X is path-connected, then we can use a path joining a to b to construct a homotopy equivalence between P(X,a,b) and St(X, a). Theorem 5.15 ([Morse34]). Suppose M is a complete, connected Riemannian maMfold, a and b are points in M and k is afield. Suppose further that Hi (P(1.1,a.b);k) {0} for infinitely many values of i. Then there are iniely many geodesics in M joining a to b. The question of the number of geodesics joining a to b in M is reduced by this theorem to a homological question. Since M is complete and connected. it is path-connected by the Hopf-Rinow theorem ([Hopf-Rinow31]) and so we study H., (1M; k) and the following result of [Seffe51]. Proposition 5.16. Suppose k is a field and there is an integer n > 2 so that Hi (X; k) = {0} for i > n and H„(X;k) {0}. Then, for any i > 0, there is an inieger j, 0 < j < n, such that Ili±i(12X; k) { 0}. That is, there does not exist a set of (n. —1) consecutive positive integers where the homology of SIX with coefficients in k is trivial in those degrees.

Hf (S2X;k)

PROOF: In the spectral sequence associated with the path-loop fibration. no nonzero classes in E,,2,„, persist to E' since P X is contractible. Suppose i is

5.2. Immediate Applications

159

the least integer for which the proposition fails and Hi(1X;k) k is a field,

{0 } .

= Iln(X; Hi (S2X; k)) H n (X; k) 0 Hi (S2X; k)

{0 } .

Since

If the proposition fails, then we have the diagram on the previous page around (n, j) in the E2 -term. In particular, there are no nonzero vector spaces that can act as a target of a differential from En2 ,i . By assumption, there are no differentials whose target is En2 ,i . Thus E° {0}, a contradiction. Proposition 5.16 together with the theorem of Morse proves that, if M is a complete, connected, Riemannian manifold such that Hi (M) {0} for at least one value of i > 1, then any pair of points in M, a, b, a b, are joined by infinitely many geodesics in M. The reader can compute H* s2n—i ; to illustrate Proposition 5.16: This computation is the result of Example 1.H. Example 5.K, in which we prove that

7ri (S 2n-1 )

is a finite group for i > 2n— 1.

Since our methods all center on homology and cohomology, we need to turn problems about homotopy groups into problems in homology. To do this we exploit the classical isomorphism of Poincaré, H1 (X) 7ri (X)/ [71 (X),7 1 (X)]. The proof of the assertion in this example is broken into steps (following 1S erre511).

STEP 1. Let X be a connected, simply-connected space. We associate to X a sequence of spaces as follows:_ X0 = X, T1 = X-0, the universal cover of X0 and X 1 = QT1 . Let T2 = X1 and X2 = QT2. By recursion, we define = X-,„ and X,2+1 = The universal cover of a space Y is simply-connected and the covering —> Y induces an isomorphism 7ri (Y) 7ri (Y) for i > 1. Among the homotopy groups of the spaces in the sequence, {X 1 j = 0, 1, ... }, we have the following relations: 7ri (X n )

7ri+) (Xn _ i ) = • • • = 7ri+, (X0 ) =

Suppose Hi (X) is finitely generated for all i, that is, X is of finite type. We proved in Example 5.A that X of finite type guarantees that QX is of finite type. We would like to extend this algebraic condition to homotopy groups. A naive argument can be made from the spaces {X1 }: If we knew that whenever Xi is of finite type, then so is Xi+1 , we would have H1 (Xi) ni (X) r i (X) and that H1 (Xi ) is finitely generated implies that 7ri + j (X) is finitely generated. However, the situation is more subtle than that. In order to make such an argument work, we need to study universal covers, which do not necessarily exist without some point set topological assumptions. EMassey67I shows that

5. The Leray-Serre Spectral Sequence I

160

a connected, locally simply-connected space has a universal cover. To prove that universal covers exist for each of the spaces X„ [Serre51] introduces the point set condition ULC (uniformement localement contractile): A space Y is ULC if there is a neighborhood U of the diagonal in Y x Y and a homotopy F : U >< I —> Y such that F(x, x, t) = x for all x Y and t I; and F(x, y, 0) = x, F(x,y,l) = y for all (x, y) E U. The relevant result is that if Y is ULC, then 1-7 exists, 17 is ULC and S2Y is ULC. Spaces that are ANRs (absolute neighborhood retracts, [Whitehead, GW781) are ULC, and this includes spaces of the homotopy type of locally finite CW-complexes. Proposition 5.17. If X is ULC, of finite type, connected and simply-connected, then 7r, (X) is finitely-generated for all i.

PROOF: To study the relationship between Y and we do not have the Lerayspectral sequence as a tool—the fibre is not connected. The tool of choice is the Cartan-Leray spectral sequence (see [Cartan481 and Theorem 8bis .9), converging to H,,(X; A) for A, an abelian group, and for which Serre

Ep2

Hp (71 (X), H-,(X - ; A)),

where we are using the homology of the group 7r1 (X) with coefficients in the ri (X) -module H(X; A). (See Chapter 8bis for definitions.) We proceed by induction. For X0 = X, since X is simply-connec ted, X = X = T1 and so X 1 = S2T1 is of finite type by the argument of Example 5.A. By induction we suppose that X,.„_ 1 is of finite type and consider T„ = [Serre511 showed that the abelian group 7ri (X,_ ) acts trivially on H(T) (since X„_ 1 is an H-space—Corollary 8bis.—)• .5The E2 -term of the Caftansimplifies for a trivial Leray spectral sequence for the covering T action: `=' Hp (71(X

• H q(Tn))

Hp(ri(Xp_i)) 0 Hq (Tp)

e Tor i (Hp_

(X,,_ )), Hq (T„)).

By induction, 7ri (X,1) is finitely generated, from which it follows that the homology groups of the group 7ri (X n _i) with coefficients in the trivial module Z, H (7r) (Xn-1)), are finitely generated (they are subquotients of the bar construction). Since the target groups Hp+q (X n _ 1) are finitely generated for all p + q and the Tor terms are finitely generated, the argument of Example 5.A applies to prove that Tri is of finite type, and hence, so is X„„ = S2Tn .

Using the Cartan-Leray spectral sequence establishes the necessary details for our naive argument, and so we have proved that 7r„„ (X) is finitely generated for all n, since 7r,„ (X) 7ri (X„,_ ).

161

5.2. Immediate Applications

If an abelian group is finitely generated, then by tensoring with Q we can determine whether the group is finite or not. In particular, if II is finitely generated, H gz Q = {0} if and only if the group H is finite. We use the spaces Xi to study 7r. (X) OE 0. In fact, we do more and establish a generalization of the Hurewicz theorem with coefficients due to [Serre51]. We write Ø for (gz in what follows. Theorem 5.18 (the rational Hurewicz theorem). If X is ULC, connected, simply-connected, and of finite type, and Hi (X; Q) = {0} for 0 < i < n, then 7r (X)®Q = {0} for 0 <j < n and7r, (X) ® Q is isomorphic to H(X; Q).

PROOF: We establish the following fact for the sequence of spaces {X } . Fact: With X as in Theorem 5.18, Hi (X 3 ;Q) = {0} if i > 0 and i +j < n = H(X; CO, if i + j = n. and, Hi (X i ; PROOF OF FACT: If j = 0, the fact follows from the identification X0 = X. Suppose it is true for 0 < m < j — 1. For j > 2, X 3 _ 1 is a loop space and so the abelian group 7r1 (Xi _1) acts trivially on H* (T3 ; Q). The Cartan-Leray spectral sequence takes the form

Ep2 ,q Hp (71 (X 3 _ 1 )) Hq (Ti ; Q)

Tor i

(Hp_ i (71 (Xj_i)),

q(Ti;Q))•

The five-term exact sequence associated to the lower left hand corner of the E2 -term (Example 1.A for homology) takes the form 0 E

d2

E 0 —> E

H,(xi_i;

E,0 O.

By induction, Hi (Xi_ ; Q) = {0} and this implies that Hi (7r) (Xi_)))

Ho (Ti; Q) {0}.

Since 7ri (X 1_ 1 ) is finitely generated—it isomorphic to H1 follows that Hi (71(X1_1)) 7r) (Xj_i) is finitely generated. Finally, Ho (Ti; Q) Q implies that 7ri(X1_i) is finite. In computing the homology of a group we can use explicit resolutions (see Chapter 8bis ), and so we can deduce (from the bar construction) that Hp (71 (X i_ i )) is finite for all p > 0 and so, when tensored with a rational vector space, gives Ep2 = {0} for p > O. The CaftanLeray spectral sequence collapses to the leftmost column and, by convergence, Hi (T3 ; Q) (Xi _ i ; Q) for all i >0. Finally, by Example 5.1 for i < 2(n — j +1) — 2, we have Hi (Xi ; Q) = (0Ti ; Q) H 1 (T1 ; Q) H 1 (X_ 1 ; Q). By the induction hypothesis, we have that Hi (X i ; Q) = {0 } for i > 0 and i + j < n and H_1 (X1 ; Q) Hn (X; Q).

162

5. The Leray-Serre Spectral Sequence I

From the fact we prove the theorem. Consider Hi (Xi) ri (X1 ) = {0 } . By the Universal For j < n— 1, j+ 1
H„ (X; Q).

STEP 2. Consider the Eilenberg-Mac Lane spaces K(:, n). We compute H* (K (Z, n); Q) using the Leray-Serre spectral sequence as follows: We have the fibrations K(Z,n —1) = S2K(7L, n)

PK(Z,

—> K(E,n)

= A(x i ), an and K (Z,1) = S'. To begin the induction, H* (K (17 , 1); exterior algebra on one generator of dimension one. The reader may now apply the argument of Example 1.H to prove, for deg xn = n,

A(x n ), =

L. Q[x„],

if n is odd, if n is even.

STEP 3. There is a mapping p: K(Z, 2n — 1) that classifies the generator of H2n -1 (S2n -1 ). As we described in §4.3, we can turn p into a libration with fibre F2 n _1. From the long exact sequence in homotopy, we know that 7ri (F2 n _ 7ri (S 271-1 ) for i > 2n— 1. We next show that 7ri (F2n _1) Q = {0 } for i > 2n — 1. Consider the mapping of fibrations: F2n-1

sar -1

s2n -1

s27 -1

IP

P

)

K(Z, 2n — 1)

This mapping induces a morphism of spectral sequences in which H* (s2n-1 ;

0)

2

,0

H* (K (E, 2n — 1); Q)

is an isomorphism and at Eco we have an isomorphism of bigraded modules because the class x2n _1 in the base is the only class of degree 2n — 1 that can survive (we know that F2n-1 is at least (2n)-connected). By the Zeeman comparison theorem (Theorem 3.26) the fibres must have isomorphic rational cohomology. Thus H* (F2n —1; Q) = {0 } . We apply Theorem 5.18 to conclude that 7ri (F2n-1) = {0 } for i > 2m — 1. Since S2n —1 is of finite type, for _ {0 } implies that 7ri (S 2n-1 ) is finite. i > 2n — 1, 7r(S21)

5.3. Appendices

163

A slightly fancier argument (outlined in the exercises) shows that 71 (S2n ) is finite for i > 2n except if i = 4n — 1. Example 5.K first appeared in Serre's thesis. This paper firmly established the place of spectral sequences in homotopy theory. The reader who is interested in the further applications of the Leray-Serre spectral sequences should skip to Chapter 6 where more structure on the spectral sequence is developed and applications given. We close this chapter with a proof of Theorems 5.1 and 5.2, establishing the homology version of the spectral sequence and the multiplicative property for the cohomology version. 5.3 Appendices

Having sketched a construction of the spectral sequence in §5.1 and discussed many of its applications, we fill in the missing steps that establish Theorem 5.1. They are: 1) setting up the apparatus of singular chains in a system of local coefficients and recognizing the -term of the spectral sequence in this context; 2) showing that d', the first differential in the spectral sequence, can be identified with the differential on these chain groups. Finally, we establish the multiplicative property of the cohomology spectral sequence (Theorem 5.2). This exposition is based primarily on the article of [Brown, E941 , which in turn is based on Serre's thesis aSerre511). In §6.3, there is a discussion of different constructions of the spectral sequence; most details of the proof due to [Dress671 are given. The reader can compare these different constructions to see what they reveal about the structure of fibrations. Finally, the reader should consult the proof of [Serre51] for the contrast between cubical singular theory and simplicial singular theory. Systems of Local Coefficients

Instances in topology where local data are patched together into global data often require more sensitive systems of coefficients in homology or cohomology. For example, at a point in an n-dimensional manifold, x E Mn , one can consider the group lin (M, M — {x}), which is infinite cyclic by excision arguments. A coherent choice of generator for each of these groups as x varies over M determines an orientation of the manifold. Obstruction theory ([S teenrod43]) presents another example of the need for more sensitive coefficients when extending cellular maps a skeleton at a time. The obstructions to lifting maps through a fibration lie in cohomology groups where the coefficients are sensitive to which cell in the base you are lifting over. A more general system of coefficients will permit a precise expression of the geometric data in the exact couple of §5.1. B Let Q(B, a, b) denote the set of paths in B joining a to b, : A continuous, A(0) = a, A(1) = b}.

164

5. The Leray-Serre Spectral Sequence I

Definition 5.19. A bundle, g, of groups over a space B is a collection of groups, {G 5 ; b E B } , togetherwith a collection of homomorphisms h[A]: Gbi 0 , one for each element in S2(B,b0,b1). This collection of hontomorphisms G5 must satisfy: (1) If c5 denotes constant path in Q(B,b,b), then h[cb] = id : Gb —> G5.

(2) If A and A' satisfy A(0) = A'(0), A(1) = A'(1), and A A' relative to the endpoints, then h[A] = h[Y]. (3) If À E 0(B, bo, bi) and Q(B,b1,b2) and A* ,u is the product path of A and then h[À = h[A] 0 h[A]: G 5 2 —› G5 0 . We also refer

g as a system of local coefficients on B.

Notice that (1) and (2) together imply that each h[A] is an isomorphism: If we denote A' (t) = A(1 - t), then A * A-1 = cbo E Q(B,bo,b0) and so h[A * A-1 ] = h[A] o h[A'] = id. Categorically speaking, a bundle of groups is a contravariant functor from the fundamental groupoid of the space B (the category with objects points in B and morphisms homotopy classes of paths between points) to the category of groups. EXAMPLES: (a) Given any group G there is the trivial bundle of groups over B, also denoted by G, with Gb = G for all b in B and h[A] = id for all A in 0(B, bo, bi)•

(b) Let H denote the bundle of fundamental groups over B, a path-connected space. Here Gb = (B, b) for each b in B. If A E Q(B b0, b,), then A determines an isomorphism h [A] : 7ri (B , 1) -> 7r1 (B, bo) by [a] [A' *a* A]. The properties of the * operation imply that we have a bundle of groups. E->.13 with B path-connected. For a (c) Suppose we have a fibration F ring R we can form a bundle of R-modules, 'H,.„(F; R), as follows: Define ET be a lifting function Gb = Hri (Fb; R) where Fb = 7 -1 (b). Let A: Qp for 7r (see §4.3) and suppose given A: I -> B with A(0) = ho and A(1) = Then define (DA : F),(1) FA(0) by (1) ),(x) = A(A -1 , x)(1). We let h[A] = 4), \*: Hri (Fbi ; R)

Hri (Fbo ; R).

By the properties of a lifting function, this is a bundle of R-modules. It is this bundle of groups that plays the key role in this chapter. (d)If we fix a basepoint * in B and let G be a group, then any representation, p: 71-1 (B, *) Aut(G), gives rise to a bundle of groups, C, over B. Let Gb = G, for each b in B; choose some [As] E 71(B, b, *) and let h[Ab] = id. Then, for any [p] E 71(B, bo, bi) let h[p] = p[Ab-01 * Ab i ]. This construction yields a bundle of groups associated to the representation p. Conversely, a bundle of groups gives rise to a representation p' : 74(B , b) Aut (G b) for b in B.

165

5.3. Appendices

(e) Given a path-connected n-dimensional manifold M, we can let G s = H„ ( AI , M — {x}) for all z E M. Via the topology of the manifold, we get a bundle of groups with which one can study the notion of orientation of the

manifold. (f) A sheaf of groups over a space B induces a bundle of groups on B, if B is a reasonable space (for example, a CW-complex). For the relevant definitions, the reader can consult the books of [Godement571 or [Wamer711. A morphism of bundles of groups, E: Ç i g2, is a collection of homomorphisms (Gi)b —> (G2)b, for each b in B, satisfying the property that the diagram hi[A] (GrObo (G1)b. Ebi (C2)b.

Ebo (G2)b0

commutes for all [A] in ri (B, b0 , b1 ). A morphism of bundles of groups is a natural transformation of contravariant functors on the fundamental groupoid. (g) Suppose we have a fibre-preserving map of fibrations, a commutative diagram

J: E

, that is

Then J induces a morphism of bundles 7-tn (F; R) —> (F' ; R). A morphism of bundles, E: gi —> g2 is an isomorphism if each Eb is an isomorphism. We call a bundle of groups G simple if there is an isomorphism E: g G, the trivial bundle of groups for some group G. The topology of B can sometimes determine a bundle of groups.

Proposition 5.20. If B is path-connected and simply-connected and bundle of groups on B, then g is simple.

g is a

PROOF: Let b be a point in B and G = Gb. By assumption 7To (S2(B ,b, x)) contains only one homotopy class of paths. If we let [A] denote this class and G is an isomorphism. = h[A], then E: g

Fix a bundle of abelian groups g over a space B. Let C. (B) denote the singular chains on B. If AP denotes the standard p-simplex in RP+1 , then let

166

5. The Leray-Serre Spectral Sequence I

vo denote the leading vertex of AP, that is, the point (1, 0,

, 0). Define the

set of singular p-chains with coefficients in the bundle g by Cp(B; g) ={finite formal sums

:

AP

Ei

0 T

where, for each i,

B is continuous and g C GT(vo)}.

The singular p-cochains with coefficients in the bundle of groups g are defined by CP (B; g) = {functions f c(B) ' LLB f(T:AP —>I3) c ai( o )}. Gb1 We describe the differential on C. (B; g). The usual simplicial boundary operator behaves as follows with respect to the leading vertex (see §4.2 for a reminder

of the formulas): (0,T)(110 ) =

T(vo), if i 0, T(v i ), if i = O.

Since the coefficients on a given simplex depend on the group in the bundle associated to the image of the leading vertex, a change of leading vertex must be B is ap-simplex, let TL: [0, id B entered into our local data. If T: AP be defined (t) = T(tvo (1 — t)vj )

be the path joining T(v i ) to T(vo ). The boundary homomorphism ah On C.(B; g) is defined on the basis by Oh(g

Lemma 5.21.

T) = h[T P](g) 0 80T +

(-1)i g ai T.

ah Oah = 0.

PROOF: From the definition of the ai we have (aiT)E2 = TI°, if i > 1 and (a0T)pg TL ; now compute. (aiT)P,„' We define the homology of B with local coefficients in G to be H.(B;g)=

Homology with local coefficients generalizes ordinary homology. Proposition 5.22. If the bundle of abelian groups Ç is simple, then the local coefficient system may be taken to be constant, that is, H.(B;g) H.(B ; G) where G = Gx for any x in B.

PROOF: Let E : g G be an isomorphism of the bundles of groups. Since the isomorphism commutes with the homomorphisms h[A], E induces an isomorphism of chain complexes, E : (B ; g) c* (B ; G). In the case of a trivial bundle of groups, the expression given for Oh reduces to the ordinary boundary operator and so H(C.(B;g), ah) gives ordinary homology.

5.3. Appendices

167

Proof of the Main Theorem

Our strategy in proving Theorem 5.1 is to follow a proof of the EilenbergZilber theorem, but for a twisted product. Recall that there are mappings AW : C„ (X x Y) <

> C. (X) (DC, (Y) : EZ,

satisfying (1) AW oEZ = id on Cp (X) 0 C, (Y); (2) there is a chain homotopy DEz: Cr (X X Y) —> C i.±1 (X x Y) such that EZ 0 — id = aDEz + DEza. If we filter Cr (X x Y) by Fp =im(EZ: Cs (X) ,s
e Cr_ s (Y) —>

Cr. (X x Y)),

the resulting spectral sequence leads to a proof of the Eilenberg-Zilber theorem, if we use homology with coefficients in a field, and to the Kiinneth theorem more generally. E Suppose F B is a fibration with F and B connected spaces. ReWE where U„ = call that 7 comes equipped with a lifting function, A: U„ {(A, e) A: [0,1] B, e E E, and A(0) = 7r(e)}, WE = map([0, 1], E), and A has the properties; 7 0 AO, e) = A and A(A, e) (0) = e (§4.3). Our goal is to analyze C.(E) in terms of C.(B) and C. (F) by introducing analogues of AW and EZ. We assume throughout that we have normalized singular chains, that is, degenerate singular simplices are set equal to zero. We first introduce some combinatorial structure available in the setting of simplicial singular chains. Let A [p]. denote the combinatorial p-simplex,

A [P]n = {(io,i), • • • , i n ) i

E {04,— ,p}

and io <j 1 <

< 2;0.

Given (i0, . , in) E A [p] n we can define a mapping An —> AP by requiring that the j th vertex of An map to the ii th vertex of AP and extending linearly to the rest of A. We denote this mapping by (i0, i1,... , in ): An —> AP. The construction leads to a pairing C2 (X) x A[p],

C,(X)

given on the basis by

(T:

—> X)

T

T >X. , i„): An (i0".. 'ia) >AP —

An example of the use of this pairing is the classical Alexander-Whitney mapping, AW: C. (X x Y) —> C. (X) 0 C. (Y). In this notation we have (T) =

n+q (prii)(0,1, 'n0 =

,n)

(pr2 T)(n,n + 1, . . . ,p + q),

168

5. The Leray-Serre Spectral Sequence I

where pr i denotes projection onto the 2 th factor. We next introduce the analogue of (X) 0 C(Y). By Cp (B;Cq (Fb )) we mean finiie sums of tensor products, Ue V, where V: A P -> B is a singular p-simplex on B and U: Ag —> Fv (p0 ) = 7 -1 (17(220)), a singular q-simplex on the fibre over V (vo ), the image of the leading vertex of V in B. This is not exactly the group of p-chains with local coefficients since we have not introduced the homomorphisms required for such a system of groups. However, these groups will lead eventually to chains with local coefficients in '11q (F). We construct the analogue of EZ based on an operation on Cp (B;C,(Fb)). Suppose U 0V is a generator where V: AP —> B and U: Ag —> Fv (r0). Let v and s denote points in AP and denote the straight line path joining v to s by [0,11-4 AP, L vs (t) = ts + (1— t)v. We define U#V : Ag x AP- E by U#V(si,s2) = A(V o L%,U(s i ))(1).

Since Vo Lus20 (0) = V(vo) = 7rU(si), U#V is well-defined. Notice, from the properties of lifting functions, that 7r(U#V)(si, 52) = V(s2). Furthermore, if a E A [q] q, and T E A [p]', and r(vo) = vo, then x T) = (Uo-)#(Vr).

We can define the Eilenberg-Zilber map for 7F as : Cp (B;Cq (Fb)) 115(U 0V) = (U#V).(EZ((0,1,.

Cp+,(E) ,q) (0,1, ... ,p))),

that is, 1/) takes U 0 V to the image of the chain (0, 1, under the composite C,(A[q].)

O Cp (A[p].)E>Cp+ ,(A[q]. x A[p].)

, q)

(0, 1, . ,p)

(U#V).

Cp+q(A q X ,6.13 )

>Cp+q(E).

The next ingredient in the proof is the filtration. Let Fp C,(E) be the subgroup of Cr (E) generated by T: Ar -> E, such that 7rT = S(io, , it), for r < p and S: AP —> B. This is an increasing filtration with FrCr (E) = Cr (E).

Lemma 5.23. The hornomorphisin I,b: Cp (B;Cq (Fb)) —> the following properties:

(1) 1P(Cp (B;Cq (Fb))) is contained in FpCp±q(E). (2) 01P =11)(OF 0 1) mod F_1.

Cp+q(E) satisfies

5.3.

169

Appendices

PROOF: From the properties of lifting functions we have the commutative diagram: Aq x AP u4v E Pr2

AP

, q) 0 (0, 1,

7T* (U#V) * (EZ((0, 1,

= V* (EZ((0, 1,

>

V

It follows that

B.

, p))

, q) 0 (0,1, • ,P)) =

V (io, • • • ,ip+q)

where 4 + , < p for all summands. To prove (2) we observe that the ordinary singular differential can be written as follows: For W E Cr (x),

o(W) = >7:_0 (-1)(w) =

—1)'W(0, 1,...

,r).

Since the singular differentials commute with all the maps that make up the composite that defines //), we have 011)(LT V) = (U#V) * (EZ(3((0, = (U#V)(EZ

(

, q) (0,

(0,

(—ni+q (0, .

,

,p))))

, q) 0 (0,... ,p)

, q) 0 (0,

,3,

,P)))

= ((aFu)#v)* (ati, an) + (-1 )q (u#(.9B1 7))* 11)(aF O 1)(u- 0 V) modulo Fp _ 1 (7, +,(E),

Ty))

where this last equality follows because OBV determines a chain in filtration P — 1.

Next up is the analogue of the Alexander-Whitney map: On generators T of FpC p+ g (E) let 0(T) = T(0, . , q) 7T(q,

, p q).

If 0(T) 0, then 7T(q, ,p q) is a nondegenerate singular p-chain on B. B, , p q) where S: Since 7 T(q, • • • , ip+q)(q, q) = S (io, we find that this is nondegenerate only when ip+ q ) = (0,

, 0, 1, ...

It follows that 7rT(0, , q) = 7rT(0, , 0) and so the q-simplex T(0,... , q) is a mapping Aq F„T(q,„. ,p+g)(vo)• Thus 0: FpCp+q(E) CP(B ; Cq(Fb)) •

170

5. The Leray-Serre Spectral Sequence I

Lemma 5.24. The homornorphism the following properties:

0:

Fp Cp±g (E) —> C,(B; Cg (Fb)) satisfies

(1) (p = id. (2) There is a chain hoinotopy D: FpCp+ q(E) FpCp±q+1(E) with OD DO = — id. (3) 0(Fp_ i Cp±g (E)) = 0. And so 0 induce s a hornomorphism 0° : El° ,q —>

Cp (B; C (Fb)).

PROOF: To establish (1) we observe that we can write EZ((0,

, q) 0 (0,

= ((0,1,

where Ti" (0 cr" = (0, 0,

,p))

,q,q,... , q) x (0,0, ... ,0,1,

p) is degenerate. If we write o-' = (0, 1, ,p), it follows that

= =

Since (0,

, p)(0 ,

, q)

(0,

and

,q) = (0, 0,

= A(V(0,

Lr(0,

= A(V o I4uo , U(0,

,p))))

, q) O Tr (U #V ).(o -' , cr")(q,

#V ).(cr t , a- NO, , 0,1, .

=

, q, q, . . . , q)

, 0,1, .

V) = 0((U#V).(EZ((0,

(/))/)(U

,p))

, 0),

,p q).

we get

, q)(—))(1) 0 V

, q)(—))(1) 0 V

V.

Here we have used the property of the lifting function that constant paths lift to constant paths. To establish the second assertion, we introduce another operation on singular simplices on E. If T: q —> E is a singular simplex on E, then define /: Aq x Aq —> E by s2) = A(71-T 0 Lss 2i ,T(s ) ))(1).

The relevant properties of Î are that 7rT (s i , 89) = 7rT (sr)), that T(8 1 ), and the following formula:

Î((0,... , q) x (q, . . . , p

q)) = T (0, . . . , q)#7rT (q,

(81, Si) =

,p +q),

which can be seen by computing T(0,... q)#7T(q,

+ q)(81, s2) , p q) 2 , T (0 ,

= A(7rT o L (q,

=

• q)(s 0)(1)

((0, . . . , q) x (q, . . . ,p + q))(si, 82).

171

5.3. Appendices

Using this new operation we define the chain homotopy D: Cp+q (E) —> Cp+, +i(E) by D(T) = (l'),:(DEz((0,

,p + q) x (O,... ,p + q))).

Because aDEz + DEZa = EZ o AW — id, we compute aD(T) = ,p + q) x (0, ,p + q))) = (f)(ODEz((0, ,p + q) x (0, ,p q))) = (f).(—DEz0((0,... ,p+ q) x (0, ,p + q)) + EZ o AW((0,... ,p +q) x ( 0, ,p + q)) —(0,... ,p+q)x (0,... ,p+q)) = —D (0 T) + (i')„(EZ o AW)((0, —

,p + q) x (0,

The last term in the list is T as it is comes down to showing that (i)(EZ oAW)((0,

,p + q) x ( 0, • • • ,p + q)) ,p +

on diagonal elements. Thus the assertion

,p + (4) x (0.... ,p

q)) = 01)(T).

To establish this, chase through the commutative diagram: cp-kg Cr X

(

+q)

al

cp+q (AP+q A P+q)

Cp±q(E)

10

AW

E

on(AP±q)

cp+q_.(AP±q) T(0,...

C (B. Cg(Fb)) P

EZ

cp ±q (AP -Fq

AP +q)

Cp±q(E)

(t). The top triangle commutes because we have / on diagonal elements; the middle

square is commutative by the definition of 0, and the bottom square commutes because PO, , x (q, ,p + q)) = T(0, , q)#7rT(q, ,p + q). Thus OD + DO = — id. We next show that 0 induces a mapping on Ep° = Fp I Fp-1. Suppose U Fp _ i Cp+,(E). Then 0(U) — U(0, , 7U(q, ,p + q) and 7VU = S(io, • • • • ip+q) with ip±q < p — 1. But (io, • • • ip+q)(q, • • • 1P ± q) is always degenerate when ip+q < p —1. Thus 0 induces a mapping,

0° :

—> Cp (B; Cq(Fb)).

5. The Leray-Serre Spectral Sequence I

172

Lemma 5.25. 00(T) = (OF 0 1)(0(T)), that is, 0° : Ep( —> Cp (B;C,(Fb)) is a chain mapping. PROOF: We compute directly: P+9

T) =

z=o p+q

, 2,

(-1)i [T(0,

.

,p + q))

, . . . ,p +

i=o 7T(0,

, q — 1)

,p + q)(q — 1, . . . ,p + q — 1)].

It follows from the definition of the pairing that

L (0,

,

,p + q)(0,

Since 7rT = 7rT(0, 7rT(q

for i < q,

,p+ q) (q , (q — 1, q — 1, q, . . .

, 0, 1,

,

— 1, q, . . . , q + ,

for i > q. , q), for i < q.

,p + q)

, q — 1) = (0,

,

we know that, for j > 0, ,p + q) =

,

p),

which is degenerate. Our formula becomes (-1)i [T(0, i-0 = (ap 01)(0(T)).

, z,

0(aT) =

7rT(q,

Thus we have the homomorphism = H (0°) :

Epl, —> Cp

,p + q)]

(B;li g (F)).

So far we have proven that there is a chain equivalence between Ep° q and Cp(B; Cg (F)). To finish the proof of Theorem 5.1, we need to show that we

have a commutative diagram, 13,(1

(5.26)

çb1-11(çb°) CP (B.

d1

ElP — 1- ,q

(F)) (- 1 )qah

4,1

C p 1

(B

To do this we introduce another chain homotopy, this time D2: Cp (B;7-1 q (F))

Fp _ i Cp±q(E).

To define the chain homotopy we require one more operation derived from the lifting function. Suppose given V: AP —> B and U: Ag —> Fv ( o ). Define U * V : Ag X AP -1 X A' —> E by * V (si, s2,t) = A(V o

oL

,

U(si))(t))(1).

5.3. Appendices

Lemma 5.27. For singular simplices V: have the following

173

AP

B and U: Ag —> Fv(v.) we

(1) U *V(s i , 82,0) — U#V(si, (1,—. '0 82)).

(2) U * V(s i , 82,1) = (Dvozzo* (U)#(0oV) (81, 82). (3) 7r(U * V) (81 82, = (ao V)(82)• 1

PROOF: We leave (1) and (3) for the reader and establish (2). (Recall (DA * is defined on p.164.) We observe first that (a0v)(to, • • •

, t_ 1 ) — v(0, to,

and so (a0V) o 420 = V o

•••

, t_ 1 ) — 17(1, • • • ,P)(to, • • •

'P)(s2) .

V(si, s2, 1) = A(V o L

,tp-1),

It follows that 0 L%, (s i ))(1))(1)

'13)(82) ,

= A((017 ) o 42° ,4) V o 1:4*(U) ( 81 )) ( 1 ) = V oL'4*(U)#(a0v)(si, s2).

Define the desired chain homotopy by D2(U0V) = (-1)P-Fg(f*V) * (EZ(EZ((0,

Lemma 5.28.

, q)®(0,

,p-1))®(0,1))).

D2(Cp(B;Cg(Fb))) c Fp _ i Cp±g(E) and

aO(U- ® V) - '0(0,1)-

+ (-1N(4)17014 *(u) ® NV)) (

(-1)4+4)(u ai -v) + (- 1)gaD2(u + H1rD2 ((apu) +

v)

where IV is a sum of elements from Fp _iCp±g_i(E). PROOF:

By Lemma 5.27(3), we get the commutative diagram Cp+g (Aq x pr2

AP -1 x

I

Cp±q(AP-1 )

(UV),

Cp+q(E) ir *

(a0 1.7),

Cp±q(B).

Since the simplices mapping into the second factor of T/32 (U , p — 1), D2 takes its image in filtration p — 1.

EZ(0,

V) come from

174

5. The Leray-Serre Spectral Sequence I

We compute directly al,b(U 0V) = a(U#V) * (EZ((0,... , 0 (0, ... ,p)))

, 0 (0, . ,p)))) 3 0V) + (-1)q (U #V) * (EZ((0 , = 11)(( FU) =

(u#v),.(Ezpo, =1

, q) 0 (1,

,p)))

(-1) i± q1P(U (aiV)).

By examining the image of the Eilenberg-Zilber map EZ on nondegenerate simplices one establishes that (U#V) * (EZ((0 ,

, 0 (1, . . . ,p))) = (Er * V)(EZ(EZ((0,

q)

0

,p - 1)) 0(0))).

Consider the chain homotopy: OD2(U 0V) = (-1)13+q0((tI *V)(EZ(EZ((0,... ,q)0 (0, —D 2 ((aF U) o V)

,p — 1)) 0 (0 ,1))))

(-1)P-Eq+1 V (U * V) * (EZ(EZ((0,.. , q) 0 (0, , -,q) (0, + (U * V) * (EZ((0 , , p — 1), 0)) — (U * V) * (EZ((0,

,q) 0 (0, .

,p — 1)) o (0,1)))

, p — 1) ,1)) .

Let W denote the big sum with terms coming from EZ applied to a product involving (0,... , ,p — 1). These classes are all of filtration p — 1. By Lemma 5.27, the rest of the morass becomes

3D2 (u v) + D2 ((aFE) v) = w + (-1) qao(u- (-1)q((3pu)g, v -) (-1)L ( u- 0 aiv) — ( 4)17 oL-0 JO 0 (00v)). For filtration reasons we have 0(W) = 0(aD2(u 0v)) =

o(D2(aFu v)) = 0.

If U represents a class in H,(Fv ( u0), then we can apply 0 to get (-1)q030(U 0 V) = 00( 3h(tf 0 V)).

This establishes the commutativity of (5.26), and we have finished the proof of Theorem 5.1 for pairs (B4O).

5.3. Appendices

Multiplicative Matters

175

g

We close the chapter with a proof of Theorem 5.2. Duality allows us to identify the E2 -term as H* (B; 7L* (F)). The dual filtration on cochains is given by FPCP+q (E) = If E CP+q (E) f (T) = 0 for all T E Fp—iCp+q(E)}.

Thus we only need to establish that we have a spectral sequence of algebras. For any space X, suppose f E CP (X) and f' E CP' (X). We define the cup-product f f' E CP+11 (X) and the cupi -product f f' E CP ±P '-1 (X) as follows: If T E Cp+p, (X) and U E Cp+p, _1(X), then (f

and (f

n(T) = f (T( 0 , • • • ,P))RT(P, • • • ,P + P')),

f')(U)= f (U (0,

+ p',... ,p + p' — 1)) f (U(i,

+ p')).

The cupi -product plays a key role in proving the graded commutativity of the cup-product as the following formula shows: 6 (f

f') = f

fl-[( 6f)

f']-(- 1 )P[f -1 (6P]-

There is a subtlety to the duality when we use local coefficients: An element of CP (B; Cq (Fb)) is a mapping of the form f: C(B) Hom(Cq (Fb), E). If we employ the Hom-Tensor interchange, then we have a linear mapping f: Cq (Fb) Cp (B) E. However, the differential must satisfy (6 f ,U

=

0(U e

Since 0 is a derivation with respect to the tensor product, the pairing on the right is with (au) + (-1) q ti (Dv). To accommodate the cross term in a different bidegree we represent a class u E HP (B;1-0 (F)) as a pair {f, y} with f: G(B) —> Hom(C,(Fb),Z) and g : c +1 (B) —> Hom(C,_, (Fb), E)• This pair satisfies, for T E Cp±i(B), U E Cq (Fb ), f , TWI) = , 01)(U) — , (0U).

If u = {f, g} is a cohomology class, then 6 f (T) = 0 and f(DT) = g(T) o as elements of Cq(Fb). Given f E C" (B; Cq (Fb)) and f' E CP' (B; Cq' (Fb)) we define a new cochain D(f, f') E CP±P ++'1(E)by the following recipe: Let T E Cp+p, +q+q, _1(E). Form two new cochains, given by E C(E)

U' E Cq, (E)

(7T(p + q +

f (7U)(T (0,

, q))

—1,... ,p + p' + q + —

Form their cupl-product and apply it to T (q, . . . ,p+q+q' —1). This procedure defines DU, , f')(T).

176

5. The Leray-Serre Spectral Sequence I

Lemma 5.29. D (f , f') has filtration p + p' when f

G CP(B;C (F))

and

CP' (B;Cq' (F5)). Furthermore, we have D(f f i ) =

*(f)

(ie') ( -1-) Pq' (f f') ±D(Si f, f') ± D(62f, D(f, 6

where 6 1 f (U)(S) = f(aU)(S), 62f (U)(S) = * f T).

PROOF: Let U

E -Fp+p'-1Cp+p'+q+

7rU = 7U(0,

D(f, 62f),

puos)

and (f,0T) =

- (E). Then we can write ,p p' — 1).

, 0, 1,

To apply D(f, , f') to U we form a certain cupi-product and evaluate oil the chain U(q,. , p + q+q'). However, 7I- U(q, , p + q+ q') is degenerate and CP+P' +7+q' so D(f, f')(U) = 0. Therefore, D(f, f') E Suppose V E C p+p, ±q+ (E) . We compute (6D(f, f'))(V) which is made up of cup products and cup) -products evaluated on expressions involving V (q, . . . ,p + q+ q'). Using the defining property of the cupi-product, we can examine each summand in turn: (f

f i)(V(q, • • ,p+ q + q')) = f (7rV (q, ,p + q + qi)(0, , q)) (V (0, . , q)).

fq (7V(p+ + q% • • ,13 + + q + qi))( 17 (q, • • 43 + q + q')(P, • • ,13 + q')) = f (7rV (q, ,p + q))(V (0, , (7rV(p + q + ,

, p + + q + q'))(V(p+ q,

, p + q + q' ))

=(f• 0*(v(0 , • • • , P+ q))) • (fi ,0*(V(P+ q, • • • ,P + + q+ qi))) ( * (f)

The sign (-1)Pq' in front of the next summand comes from the formula for the differential applied to a cup l -product and the fact that we are using a p-cochain and a q'-cochain. (f'

f)(1/ (q, ,p + q + q')) = f' (7TV(p+ q + q' , ,p + p' + q + q'))(V (q, ,p + q +qi )(0, . . , q')). f (7rV

, p + q + q')(q,

= f (7rV (q + q' ,

, p + q'))(V (0,

, q))

, p + q + q'))(V (0, . , q)).

(7rV(p + q + q' ,

,p + + q + qi))(V (q,

, q + q'))

,p+q) , V (q,

TV (q+ q' , ,p + q + q'))• , q + 47') 7rV + q + , ,p + + q +

=f

f' , V (0, • ,q + q')

71/(q

=(f

f',0*(T)) = (0* (f

n)(T).

q',

,p + + q + q'))

Exercises

177

We are looking to establish a map on the E2-term and so we may think of our class f as representing u E HP(B; Hq (F)), that is, it is part of a pair {f, g } with 6 f (U)(S) = f (U)(S) + (5 2f (U)(S). ((61)

f i )(V(q , • • •

=

qi ))

113

17 (q, • • •

43 +

OA • • • ,i,

qi , • • • 43 + qi ))*

f', 17 (q, • • • ,P+ + qi )( 2:, • • • , i+qT•

The cochain (6 f ,V (q, , p + q + q/ )(0, + , . . . ,p+ q')) applied to a q-chain on F gives something of the form (f, au)(s) + 0, U) (as). Since f is a cocycle, the first part of this sum vanishes and so (S f ,U)(S) = 0, U) (OS). Thus (6 f) —1 f' has filtration p + +1 and it vanishes when we take the associated graded module. At the E2 -term, 0, U) is a cocycle and so all terms involving 62 vanish. Since these remarks apply to all the terms coming from f' and f —1(6 f), we get in the E2 -term (6f) (0'2i) — (0'21') = ( -1 )Pg' 0'2(f and we have proved Theorem 5.2.

f')

Exercises 5.1. Prove that the cellular fi ltration that gives rise to the Leray-Serre spectral sequence is exhaustive. 5.2. Derive the Wang long exact sequence for a fibration F E B with B a simply-connected homology n-sphere, n> 2, and F, path-connected:

H k( E; R )

Hk ( F; R) L

(F; R)

Hk+1 (E; R)

Show further that 9 is a derivation, if n is even and when n is odd, O is an antiderivation (0(x ----- y) = 0(x) y+x 0(y)). 5.3. Establish the Serre exact sequence (Example 5.D) in the case of homology. 5.4. Prove Proposition 5.5. 5.5. Theorem 5.1 can be extended to librations over a CW-pair (B, A). Suppose F E B is a fibration, A C B, EA C E, and p-1 (A) = EA, that is, we have a libration of pairs, F

(E , Ei) L (B A).

Sketch a proof of the homology Leray-Serre spectral sequence for pairs for which the E 2 -term takes the form Ep2,

Hp (B , A; 'H,(F ; G)),

and the spectral sequence converges to H,,(E, EA; G).

178

5. The Leray-Serre Spectral Sequence 1

5.6. There is another notion of fibration for pairs: Let F E L B is a fibration, F0 C F, and E0 C E, such that plEo : E0 B is a fibration with fibre F0. That is, we have a libration of pairs

(F, Fo )

(E, E) 4 2 B.

Derive the homology Leray Serre spectral sequence in this case for which Ep2, Hp (B;1-tg (F, Fo; G)), converging to H.(E, E0; G). -

5.7. Prove Corollary 5.8. 5.8. Show that F is totally homologous to zero with respect to k if and only if the following holds for the Poincaré series with respect to k:

P(E,t) = P(B,t) x P(F,t).

5.9. Determine the nonzero differential in Hirsch's example of a fibration F

E

B with p* injective but for which the cohomology Leray-Serre spectral sequence does not collapse. (Another example is given by the free loop space associated to a space X: Let LX = map(S 1 , X), the space of unbased loops on X. Evaluation at 1 G Sl gives a fibration S2X

LX

evi

) X. Sending x E X to the constant loop at x gives a section of evi, and so evt is injective. However, the associated Leray-Serre spectral sequence need not collapse (see [Smith, L811 and [McC1eary90] for explicit examples). -

SU(n)/SU(n — k) where SU(n — k) C SU(n) is included as the lower right hand corner of an otherwise n x n identity matrix. 5.10. Show that 1,7),(01 )

5.11. Prove Theorem 5.13. 5.12. Show that Vk (Rn ) is (n — 0-connected. 5.13. The symplectic groups, Sp(n), are the analogues of the special orthogonal or unitary groups over the quaternions. There are fibrations defined analogously, Ss , compute H*(Sp(n)), for all Sp(r/ — 1) Sp(n) —> S 471-3 . From Sp(1) n. The quaternionic Stiefel manifolds Vk(E-r) are defined similarly to the real and complex Stiefel manifolds. Compute H* (Vk (IEI)). 5.14. Compute H* (SO (n); Fp), for p, an odd prime. 5.15. The exceptional Lie group, G2, is represented by the group of automorphisms of the Cayley numbers, K. (See [Whitehead, GW78, Appendix A] for a discussion of this division algebra.) There is a fibration that results from this observation; S3 G2 —> V2 (Th.7 ). Compute H* (G2; 72) • 5.16. Show that Sn is ULC.

179

Exercises

V2 (Rn+1 ) —> Sn be the libration sending the orthonormal 2frame (2/1 , 2/2 ) to 1/ 1 . Show that this fibration has no section if n is even. Modify the argument of Example 5.K, using this fibration, to show that 7ri (S 2m ) is finite, except when j = 2m or i = 4m — 1. In particular, show that )74,),_1(52m ) finite. (Hint: Compute 11-* (V2 (Rn +1 ) ; Q) and apply the rational Hurewicz theorem. Then

5.17. Let S n—

e

study the long exact sequence of homotopy groups associated to the libration.)

6 The Leray-Serre Spectral Sequence II the behavior of this spectral sequence is a bit like an Elizabethan drama, full of action, in which the business of each character is to kill at least one other character, so that at the end of the play one has the stage strewn with corpses and only one actor left alive (namely the one who has to speak the last few lines)." ...

. ..

J. E Adams

We begin with a classical computation that demonstrates further the potential of the Leray-Serre spectral sequence in homotopy theory; we also introduce several of the notions that will play as leitmotifs through this chapter. Our first goal is to prove the following result. Theorem 6.1. 74(S2 )

Though this fact may be obtained by more elementary means ([Whitehead, GW501), the proof here is based on a general method (albeit limited) for investigating homotopy groups and which leads to such results as the proof of [Brown, E571 that the homotopy groups of finite complexes are finitely computable. The technique is to apply the spectral sequence to stages of the Postnikov tower of S2 (§4.3). Each stage is a pullback of the path-loop fibration over an EilenbergMac Lane space (S2 is simply-connected). We compute H* (K(7, n)) in some low dimensions in order to proceed. Further progress might be possible if more were known about H* (K n)). This problem was solved by [Cartan541. The computation of H* (K (7, n); F,), however, is simplified significantly by the presence of the Steenrod algebra. The key to the mod p computation is the transgression and its properties. The development of this feature of the Leray-Serre spectral sequence occupies the first third of the chapter and culminates in the determination of H* (K (17 Ip17 , n); 719) and H* (K n); 7p) due to [Serre531 for p = 2 and to [Cartan54] for p an odd prime. [Serre531 used these calculations to demonstrate the nonvanishing of the homotopy groups of finite complexes in infinitely many dimensions. His argument involves an ingenious use of Poincaré series and the method of 'killing homotopy groups'.

6.1.A Proof of Theorem 6.1

181

The middle third of the chapter turns the spectral sequence apparatus, together with the transgression, to computations of the cohomology of classifying spaces and characteristic classes of fibre bundles following [Bore153]. We begin with a sketch of the theory of classifying spaces and their construction. The computation of H* (BC; k) makes up a major portion of this section that closes with the relation of these computations to the notion of characteristic classes. In the final third of the chapter. we discuss some different approaches to the construction of the Leray-Serre spectral sequence. We begin with a discussion of the original construction of [Leray50] and a structure result for differentials due to [Fadell-Hurewicz58]. The next construction considered is due to [Brown, E59 1 and has its roots in the classical theorem of [Hirsch, 0 54 1 on the cochains on the total space of a fibration. [Brown, E59 1 introduced the machinery of twisting cochains, a kind of generalization of the Eilenberg-Zilber theorem, to determine how the cochains on the total space are related to those on the base and fibre. A spectral sequence arises naturally from this construction that is identifiable with the Leray-Serre spectral sequence. The other construction considered in detail in §6.3 is due to [Dress67], where a bisimplicial set is associated with a fibration. The Leray-Serre spectral sequence is derived from the double complex associated to a bisimplicial set. Finally, some results on the uniqueness of the Leray-Serre spectral sequence, due to [Barnes851, are described to close out the chapter. 6.1 A Proof of Theorem 6.1

We begin by recalling the Postnikov tower for S2 :

) K(75 (S2 ), 6)

P4S2

S2

P3 S2

k5

k4

K(74(S 2 ), 5)

j;;N K(73 (S2 ), 4)

P2 S2

Each space, Pn S2 satisfies the properties (1)

(P,52 )

f 7ri (.92),

{o},

for i < n, for i > n.

(52 ) —> 7,(PS 2 ) induces the isomorphism for i < n. (3) The mapping P,S2 Pn _ 1 S 2 is the fibration pulled back from the pathloop fibration with base K (7,2 (5'2 ), n 1) over the mapping k n . (2) f, :

182

6. The Leray-Serre Spectral Sequence II

By (1) and the fact that 7r2(S 2 ) Z (the classical theorem of Brouwer), P2S 2 has the homotopy type of the Eilenberg-Mac Lane space K(Z, 2). Furthermore, we can identify K(Z, 2) with Z P(œ). Thus we know the integral cohomology of P2S2 completely; H* (P2S2 ) Z[x2j, where a)2 has degree 2. Next observe that (2) and the Whitehead theorem imply that, for i < n, Hi (f„): Hi (S 2 ) Hi (P„S2 ) is an isomorphism, and for i = n + 1, an epimorphism. The Universal Coefficient theorem allows us to compute the low dimensional integral cohomology of P„S 2 : if i = 0, 2, fob if i= 1, 3, 4,

/7 , IP(PruS 2 )

,n + 1 .

As we work our way up the Postnikov tower, H* (13,5'2 ) can be determined, in principle, in terms of H* (P„_1 S 2 ) and H* (K (Tr „ (S2 ), n)) via the Leray-Serre spectral sequence. We record some values, in low dimensions, of H(K n)) and H* (K(qv,n)). Lemma 6.2. For iv, a finite abelian group, and n > 2, Hn (K (Tr n)) Hn+1 (K (rr , n)) and Hri (K (Tr, n)) = {0} =

iv

PROOF: By the Hurewicz theorem we know that H„_1 (K (rr ,n)) = {0} and H „(K (7r, n)) 7r. The Universal Coefficient theorem gives the short exact sequence 0

—

> Ext(H, _ 1 (K (7 ,n)),Z)

Hn (K (7r, n)) Hom(Hn (K(7r, n)), ?Z) —>

O.

Since Hom(7r, Z) is {0} for 7r, a finite group, we find Hn (K (Tr ,n)) = {0 } . We next consider the path-loop fibration K (7,n

—

1)

P K (7r , n)

K (Tr , n).

By the connectivity of K(7, n) and repeated application of the Serre exact sequence (Example 5.D), we have Hn +i (K (7, n)) Hn (K(7r, n 1))

•••

H3(K(7r, 2)).

By the fundamental theorem for finitely-generated abelian groups and the properties of Eilenberg-Mac Lane spaces, it suffices to prove the theorem for iv = E/TriE and n = 2. The short exact sequence 0 Z/mZ —> 0 gives a fibration K(Z, 2) —> K(Z, 2)K(/m, 2) (up to homotopy) realizing the short exact sequence of groups on homotopy. In low dimensions there is a single differential in the associated spectral sequence to determine: d3: H3 (K(Z/mE, 2)) —> = H2 (K(2, 2)).

183

6.1. A Proof of Theorem 6.1

If 113 (K(/m, 2)) were finite, then d3 0 and this finite group would persist to E" to give a nontrivial contribution to H3 (K (Z, 2)) = {0 } . The Serre exact sequence in homology gives the exact sequence O H3 (K (E/ME, 2) )

d3 .f/2 (K (E, 2)) —> H2 (K (E, 2)) —> H2 (K(Z/mZ, 2)) —>

O.

Thus H3 (K (747127Z, 2)) can be identified with a subgroup of Z. It follows that 4°2 Z/qZ for some q. This presents an extension problem to reconstruct H2'(K(Z, 2 ) ) Z with the data E(c)79 ZAZ, E;"°1 = {0} and Er;°0 Since this is impossible, H3 (K(74/1117 , 2)) = {O}. A second use of the Universal Coefficient theorem gives the exact sequence 0 —> Ext(H(Ke7r, 70), Z) —> Hn+1 (K ( rr , n)) Hom(Hri+ (Ke7r, n)), Z)

Since Ext (7, Z) 7T, the short exact sequence gives 7r

—40.

Hn+1 (K(7r, n)).

From the properties of Horn and Ext, we deduce that, whenever 7T is a finitely generated abelian group, Hn (K(77, n)) = Z7, and Iin+' (K(7, n)) {0} imply that 7r = Z. Consider the next fibration in the Postnikov system: K(73(S 2 ), 3)

inc )- P3S2

CP(oo).

The homology Leray-Serre spectral sequence for P3S2 diagram.

z—

P2S2 leads to

the

— — 0 — Z — 0 —

By the connectivity of the mapping f3 : S2 > P3S2 and the Whitehead theo-3 rni But this can only occur if the differential rem, 11 (p3s2) — H4 (p3 s2) _ tUjd4: Ezt 0 Z Z. 7r3 (S 2 ) 41,3 is an isomorphism. Thus 73(5' 2 ) —

184

6.

The Leray-Serre Spectral Sequence II

To obtain further homology groups of P3 S2 , we must consider H* (K (E, 3)) in a few more dimensions. The path-loop fibration over K(Z, 3) gives the following diagram for the cohomology spectral sequence in low dimensions:

Since d3 is a derivation, d3 (x 2 ) = 2xy. Furthermore, the group ? in the „ d3

d3

diagram must be part of an exact sequence, 0 —> Exy ? O. Thus ? E/2E = H6 (K (E, 3)) . Putting this further data into the cohomology Leray-Serre spectral sequence for K(E, 3) —> P3S 2 P2S2 CP(00, we see that the bottom degree class in H3 (K(E, 3)) must go to the generator of H4 (CP (oo)) in order that H3 (P.3 52 ) = {0 } . The algebra structure of the spectral sequence takes over; the next possible case of a differential must come from H6 (K (E, 3)) but such a class has nowhere to go and we conclude H6 (P35 2 )

Finally, consider the fibration K(74 (S2 ), 4) —> P4S2 P3 S2 . In the cohomology spectral sequence, we have the data E2,° E, E26 '° E/2E, and E2i ' 0 = {0 } for 2 < j < 6. This leaves Hi (K(71-4(S2 ), 4)) unknown for i = 4 and 5. Since H4 (P4S2 ) 101, it follows from the placement of holes that H4 (K (7 -4 (S2 ), 4)) = {0 } . (This is also a corollary of the argument of Example 5.K for even-dimensional spheres, that is, 74 (S2 ) is finite.) From the Universal Coefficient theorem we deduce that H5 (P4S2 ) = {0 } . It follows that d6: H5 (K(71-4 (S2 ), 4)) —> H ° (P35 2 ) is an isomorphism: H5 (K(74(S 2 ), 4)) 2Z/227_, and, by Lemma 6.2, we have proved Theorem 6.1.

In the course of the proof of Theorem 6.1, we have computed some of the groups H* (K(n . , n)). The reader is encouraged to try his or her hand at extending the computation. The p-primary components of H* (K (E, 3)) plan an important role and this reveals the difficulty of computing these groups in general. Employing the philosophy that problems, taken one prime at a

6.2.

185

The transgression

time, may be more accessible, we turn to the computation of the algebras H* (K (7, n); IFp ), for p a prime and 7r a finitely generated group. The key tool in these computations is the transgression and its algebraic realization in the Leray-Serre spectral sequence. 6.2 The transgression

The Russian term for a fibration is literally a 'twisted product' (of the fibre and base). An algebraic measure of the nontriviality of the twisting is found in the long exact sequence of homotopy groups of the fibration: * • • • —>

q (E)

q (F)

(F)

•••.

The nontriviality of 0, the connecting homomorphism, gives an indication of how the base and fibre are twisted together to form the total space. Though historically inaccurate, we follow [James611 and call a the transgression of the fibration (F, E , p, B). (See [Chern481, [Hirsch481, and [Koszu1501 for the first notions of the transgression.) To define the transgression on homology groups we consider the following commutative diagram where h: 7rg (X) —> Hq (X) denotes the Hurewicz homomorphism: 7r

7rq (F)

E)

*

71" q (B)

7rq -1(F)

Hq (E, F)

Hq _ 1 (F)

> Hg (B ,*)

> Hq _ 1 (*)

h

Hg (F)

Hq (E)

) HI (*)

Hq (B)

1* P

Here P0: (E, F) —> (B ,*) is the induced fibration of pairs. The exactness of the two bottom rows is used to define the transgression homoniorphism. Definition 6.3. Let T denote the homomorphism, r:

1 (iril p o *)

H,_ 1 (F)

la(ker po*

given by r(z) = Or + 0(ker po* ) where z G Po*), and Po*1 (.1*(z)) = r + ker po* . The homomorphism T is called the transgression.

We leave it to the reader to prove that T is well-defined. Notice that T maps a subgroup of H „(B) to a quotient of II (F). The definition of T for homology with coefficients other than E is clear from the diagram. The following is an immediate consequence of the definition.

186

6.

The Leray-Serre Spectral Sequence II

Proposition 6.4. If a class u E Hq (B) is spherical, that is, u is in the image of the Hurewicz homomorphism, then r(u) is spherical in Hg _ i (F)/a(kerpo ), that is, r(u) lies in the image of the composite mapping q_1(F) —> q -1 ( F)

I

11- q -1(F) a(ker PO*) •

The dual notion may be defined for cohomology as follows: Consider the commutative diagram with exact rows: H q –1 (*)

Hq (B , *)

Hq (B)

Hq (*)

Hg (E)

Hg (F)

6

pol

H_1(F) 6 > Hg (E, F)

o

Define T : imp) c Hq _ 1 (F)) Hq (B)Ij*(ker A) by T (Z) = j'(kerp') where z E '(imp) and /4 (r ker p'd ) = 6z. If we replace H*( ) with H* ( ; R), we get the definition of T for cohomology with coefficients in a ring R. In particular, for R = Fr), p a prime, the action of the Steenrod algebra commutes with all of the homomorphisms in the diagram. ('

(

Proposition 6.5. If is in A p, the mod p Steenrod algebra, and z, a class in (5 – '(imp) c Ha-1 (F;Fp ), then r(Oz) = 07(z).

Here equality means as cosets of ker p* in H* (B; Ip ) or as elements of imp* H* (B; ) /ker p* . It is this property of the transgression that is the key to later computations. The main theorem of this section identifies r, a feature of the homological data of the spaces in a fibration, with an algebraic feature in the Leray-Serre spectral sequence. Theorem 6.6. Given a fibration F E— >.B with base B and fibre F connected, the following hold for its associated Leray-Serre spectral sequence C (2) E610,_ 1 Hn _i(F)/a(kerpo * ), and (3) d: is the transgression. (1) 4,0

PROOF: We begin with a proof of part (1). Consider the map of fibrations (E,

129° B

(B *)

6.2. The transgression

187

By the naturality of the spectral sequence (Theorem 5.1), the inclusions induce a mapping j* : Err„,q —> Err for all r, m, q where E„.2 Hm (B; H* (F)) and Er% Hm (B ,* ; H* (F)). Observe that Er02 ,* Ho (B,* ; H* (F)) = {0}, and E Er2„ g for m > O. Since j* commutes with the differentials, j* : 4 0' 4 0 for k < n. Furthermore, Ed,* = {0} implies En' E. By Theorem 5.9 EnI 1 can be identified with im (po* : H„(E, F) —> Ha (B,*)) and so the isomorphism j* : E ,0 —> Enn ,0 determines E:,0 jz 1 (impo* ). This proves assertion (1). We next consider some important diagrams. The first displays a reformulation of the definition of the transgression: 0

> ker po*

H,(E,

PO*

10 H_ 1 (F)

;

im po

>

0

a

0 — a (ker po* )

i (F)/a(kerpo * )

) O.

The second diagram, with rows exact, is based on the naturality of the spectral

sequence (Theorem 5.9). 0

ker po*

H(E)

H,(E, F)

P* 0

PO*

1

EZ0

E;;1,,c,

H(B)

H,(B, *)

a(kerpo*)

°

H_(F)

Is

(A)

On-1 Er°°

On,n -1

E

°

0

diagram commutes for obvious reasons; however, it remains to prove that the middlemost square commute. The homomorphism denotes the surjection given by the composite Most of the

Fin–i(F) E13,n–i

>>

)>E

1

»

6.

188

The Leray-Serre Spectral Sequence II

Lemma 6.7. (A) is a commutative square.

We postpone the proof of this lemma briefly and gather its corollaries. We first prove part (2) of Theorem 6.6: Suppose u 0 is in Ha_i(F) and ((v) = O. Then i,, (u) = 0 by the commutativity of the diagram and so u is in ker i. = im a. Let w be in Hn (E, F) with 0(w) = u. By the commutativity of (A) cln(po * (w)) = O. If po.(w) 0, then po. (w) lies in ker dn = E 0 = imp* . Thus there is a v in Hn (E) with po.(j(v)) = po.(w). Let w' = w — j.(v). Then po.(w') = 0 and a(w') = d(w) — 5j * (y) = 0(w) = u. Thus ker ( = a(ker po.) and (2) follows. To derive part (3) of Theorem 6.6, return to the first important diagram after the theorem and substitute the conclusions of parts (1) and (2) to obtain the commutative diagram 0

0

kerPo.

o 0

O.

a(kerpo*)

To complete the proof of Theorem 6.6 it suffices to establish Lemma 6.7. PROOF OF LEMMA 6.7: We use the definition of the groups appearing in this part of the spectral sequence. In particular, E ,0 = Z77,0 / B rin3 1

En 0,n-1

= Zon,n—i

, 0,n-1

where = {x E Fn Cn (E) 0(x) G F0Cn _i1E1 = Cri—i(F)} B 7,1 ,73 1 = {x E FnCri(E) there is y E F2,1 _1C71+11E1 with a(y)= x} = Bn (E) = {x E Fn—iCa(E) = {x

Fo c

a(x) E Foci(E)= c,(F)} c FnCri-2(E)} = FoC1(E)

i (E) a(x)

.737j,„7„1 1 = {x E FoCri 11 0 there is y E Fn _tC(E) with a(y) = x}. -

-

At the level of the definitions, the differential dn can be written as dn (X ± Bn (E) 411,1 ) = a(x) 1407, 1 1 .

However, Ha (E, F) = Hn (C(E)/ Fo Cn (E)) and the mapping po ,. takes U E Hn (E, F) to the class u + 13n (E) +ZI, 1 ; since 0(u) E FoCn _i (E), and we have an increasing filtration, this is well-defined. Furthermore, d(u) E

6.2. The transgression

189

H_ 1 (F) is given by the class 0(u) + Bn_i(F). The mapping takes y B,.„_ 1 (F) c H_ 1 (F) to the coset y + . However, B,.„_ 1 (F) c /3 1

so this is well-defined. Finally, we chase around the square in each direction. A class u+ F0 C, (E) with 0(u) FoCp_ (E) goes by d' o po* to 0(u) + .147n 1 1 . Likewise, the same class is the image of u FoCn (E) under 0 O. For completeness we include a statement of Theorem 6.6 for cohomology. The proof is left to the reader.

Theorem 6.8. Given a fibration F E— >.B with base B and fibre F connected, the following hold for its associated cohomology Leray-Serre spectral sequence (1)

Hn (B) /ker p* ,

(2) 4,--1 6- '(imp) c (3) eln : En° '71-1

Enn '° is the transgression.

Combining Theorem 6.6, Proposition 6.5 and Theorem 5.2 we obtain the following corollary for mod p cohomology. Corollary 6.9. Suppose u in H' (F; 7p) survives to E n"' and 0 is in A p, the mod p Steenrod algebra. If 0 has degree k then dn+k(0u) = 0(dpu).

We can present Corollary 6.9 pictorially in the diagram:

The homology transgression is an example of an additive relation, a homomorphism defined on a subquotient of a module to another subquotient.

Consider the part of the long exact sequence in homology of the pair (E, F): a Po. Hq _1(F) HJE,

Hq (B,*) = Hq(B).

6.

190

The Leray-Serre Spectral Sequence II

The transgression is defined on the image of po* and it is determined up to the kernel of po* . So we have the equivalent expression r: impo*

(E, F)/ kerpo*

o

Hq _ i (F)/a(kerpo* ).

For a pointed space (X, x 0 ), we include X C CX as the subspace of classes [x,1]. For a cofibrant space, CX/X SX and the pair (CX, X) yields a similar pair of homomorphisms: a

Hq _1(X) H q (C X, X) —> Hq (S X, *).

Since the cone on X is contractible, 0 is an isomorphism and the homomorphism E = Q o 3 -1 : H,_ 1 (X) —> II,(S X) is the classical suspension isomorphism. Suppose (F, E,p, B) is a fibration and Hq (E) = Hq _ 1 (E) = {0}. Then o Hq (E, F) is an isomorphism and we can the homomorphism Hq _ 1 (F) form the composite, opposite to the transgression but in the order of the classical suspension, to produce the homology suspension a = po* 0

Hq _ 1 (F) —> Hq (B).

When E = PY in the path-loop fibration, (0Y, PY, ev i , Y), the homoHq (Y). This logy suspension is defined for all q > 0, : Hq _ (SW) Eilenberg -Mac homomorphism was introduced by Lane50] in their study of the homology of K(7, n)'s, and further developed by [Serre5 1 1 . From the Serre exact sequence for the path-loop fibration, if Y is (n — 1)-connected, then E* : Hq _ i (S2Y) —> (Y) is the inverse of the transgression and an isomorphism for q < 2n — 2. More generally, [Serre511 proved the following result. Proposition 6.10. Suppose F E—>B is a fibraiion with B path-connected and F connected. Suppose that Hq (E) Hq _1(E) = {0}. Then the homology suspension E* : Hg _ 1 (F) —> Hq (B) is well-defined and makes the following diagram commute: cri Eq0,q-1 E 0 S

Hq (B)

E„

In particular, E* has the same kernel as Hq (B).

Hq _ 1 (F)

and the same image as the inclusion

The proposition follows from Lemma 6.7. We next relate the suspension isomorphism with the homology suspension.

191

6.2. The transgression

Lemma 6.11. Let f: X —> SW. be a pointed map and adj (f): S X —> Y its adjoint. Then there is a mapping f: cx PY making the following diagram commute X > CX SX

l

5217

PY

adj(f)

Y.

PROOF (IAdams601): Recall the identity of all the maps in the diagram: If f: X —> 12Y - is written f (x) = A x , then adi(f )([x , t]) = Ax(t) = f (x)(t). The quotient map Q: CX —> SX takes [x, t] to itself when x xo and for 0 < t < 1, and [x, 0], [x, 1], and [xo , t] all go to the same class, the basepoint of SX. Since f is pointed, f (xo) = cyo , the constant loop at yo E Y, the basepoint of Y. Define f : CX —> PY by f tp(s) = f (x)(st). Notice that t])(0) = f (x) (0) = yo and f-([x,1])(s) = f (x)(s) and so the left side of the diagram commutes. Since f ([x , t]) (1) = f (x)(t) = adj(f) , tp, the right square commutes. 111 On homology we get the following diagram relating E and E.: a

Hg (C X, X)

Q * > Hq (SX) adj (f).

Hq (PY,S2Y)

evi

11q(Y)

When Y = SX and j: X —> QS X is the mapping satisfying adj(j) = id: SX SX, we obtain the commutative diagram: Hq (X)

Hq±i (SX)

By the Serre exact sequence for (S2SX,PSX, evi, SX) we know that E. is an isomorphism for q < 2n where X is (n — *connected. This implies that j. is an isomorphism in this range and the Whitehead theorem implies that j. : 7rq (X, X0) 71-q (S2SX, Sxo) is an isomorphism in this range and an epimorphism when q = 2n ± 1. Since the adjoint of the identity mapping on SX induces the suspension mapping on homotopy groups, we have proved the classical theorem:

192

6.

The Leray-Serre Spectral Sequence II

Theorem 6.12 (the Freudenthal suspension theorem). If X is (n — 1)connected, then : x o ) —> 7,+i (SX, Sx0 ) is an isomorphism for all q < 2m and an epimorphism for q = 2m ± 1. We investigate the homology suspension and its dual further in §8.2. In the next section, we investigate another way in which the Steenrod algebra appears in the workings of the transgression. Kudo's theorem

In this section we restrict our attention to mod p cohomology for p a prime. Many computations in which the transgression figures prominently involve fibrations with acyclic total space—a nontrivial transgression should be expected since a: 7n (B) (F) is an isomorphism in this case. Special attention must be given to every possibly nonzero class in the spectral sequence; in particular, the transgression can give rise to nonzero product elements. In light of this, we extend the definition of the transgression to include such classes. Definition 6.13. Suppose u E E2s ' t . We say that u is transgressive if d2 (u) = If y / Es+t+ 1 ,o d3(u) = • • • = d(u) = 0 and dt±((u) 0 in Es+t+1'° 2 t+i survives to represent dt+i(u), then we say that u transgresses to V.

Notice that it is not properly u that survives to E 1 but that u represents a class in Et's2 1 with nonzero image under dt+ 1. Elements in the subspace 6 -1 (im A) of H' (F; p ) (see §6.1) are transgressive according to this definition when their image under the cohomological transgression is nonzero. Suppose a class x E H2k (F; Fp) is transgressive and y G B-2k-ki(B ; Tp ) represents d2k+1(x). Because d2k-i1 is a derivation (Theorem 5.5), we have, for 1 < i < p, d2k-k1(x i ) = iy 0 x i-1 . However, two phenomena come into play when i = p. First, from the derivation property and the mod p coefficients, we have d2k-k1(x P) = py 0 xp -1 = O. Alternatively, xP = Pkx and so, by Corollary 6.9, if Pkx survives to E204/71 , then d2kp-ki(P k x) = Pk y. This 22kk++21,2k(p-1). leaves y 0 xP -1 a nonzero class in E 1Kudo561 determined the behavior of these elements in the spectral sequence in terms of the Steenrod algebra action on H* (B; IF). Theorem 6.14 (the Kudo transgression theorem). If a class x in E2° ' 21 B-2k ( F; r=1:,) is transgressive and x transgresses to the element represented by y in E22k-ki,o H2k-ki( B; Fp ■) , then Pkx = xP and y® xP -1 are also transgressive with d22k-k1(xP) = Pk y and d2(p_l)k-kl(Y OCP-1 ) = — ,3 Pk Y. (Here /3 denotes the rnod p Bockstein in Ar. ) IDEA OF THE PROOF: A complete proof of Kudo's theorem requires careful consideration of a construction of the Steenrod algebra action at the cochain

193

6.2. The transgression

level on the cohomology of a space. Following [Steenrod57], 11May70] gives such a construction in a very general setting. Steenrod operations are based on the Fp [/pz] -free, acyclic complex

w0

6 W1 ,

6 -wn ,

with Wn the free Tp [E/pE] -module on a single generator en . If E/pE is generated by an element a (written multiplicatively), then 6(e2i) = ( 1 ± + ".± a P 1 )e2i—i,

(5. (e2i+i) —

—1 )e2i.

The complex (W*, 6) plays a role in the cohomology of groups (see 8§bis 2). The additional structure needed to define Steenrod operations is a morphism Q of Fp ppq -complexes, where 7p [Z/pZ] acts trivially on the target C*, and permutes the factors of (C*)®P, 0: W-* ø (C*) (2)P —> C.

The mappings Di (a) = a®P) are identified with the cocycles representing the Steenrod operations applied to [a] E H(C*). The structure map extends the p-fold iterated product on co 0 (C*)®P. 11May70, pp. 166-7 1 presents an algorithm for constructing cochain representatives for these operations. We can apply the construction to C*(E; 729) and to representatives of transgressive elements. As we saw in the proof of Theorem 6.6, being transgressive is a condition on filtration degrees. [May70] shows that if x E E2° ;c2k 22 kk+ , then d2 (p 1)k +1(Y 0 x 19-1 ) has a cochain repretransgresses to y E E"° sentative 0e(e o xP -1 y) that represents —0Pk(y) modulo terms of higher filtration. The key to the proof is the careful construction of cochain level operations along with the identification of the transgression with a condition on filtration degree. May's technical but elegant proof applies more generally to other spectral sequences.

6.

194

The Leray-Serre Spectral Sequence II

Kudo's theorem shows that the elements of the form y 0 xP -1- and the Steenrod algebra action play crucial roles in describing the transgression. We make two further comments along these lines. In the calculation of H. (K (7r, n);7,) and H* (K , n); Fp ) as algebras, [Cartan54] had already noticed the role of the classes y 0 xP -1 . Cartan introduced the idea of an acyclic construction of algebras over Fp : a triple, (A, N, M), with A, N, and M algebras where M possesses a differential and H(M, dm) M is a twisted version of A® N with the differential on M a morphism A —> N of degree ±1 (depending on whether we are computing the homology or cohomology). The motivating idea is to obtain a version of the chains on the total space of a fibration from the chains on the base and fibre when the total space is acyclic. If one ignores the filtration in the Leray-Serre spectral sequence and considers the total complex,

(total E) =

Er' s

r-ks=q '

r-ks=q

Hr (B;

)

H (F; Fp )

along with all of the differentials simultaneously, then this gives a suggestive image for the idea of a construction. The technical details are given by [Cartan541 (also see [Moore761 and [Stasheff871). For (A, N, M), an acyclic, multiplicative, graded-commutative construction over 7p , Cartan defined an additive function for q> 0, t//: p (A2,)

#

H2pq+2 (N)

where p (A2g ) = {a E A2 q such that da = 0 and aP = in,. The function tp determines the reduction in N of a class y in M with dy = 0-1 x and dx = a. Under further conditions, satisfied in the problem of computing H. (K(7r, n); , the mapping 't/) determines a homomorphism (i0 : H2q (A)

H2pq+2(N)

called the transpotence. In the case of Eilenberg-Mac Lane spaces, the transpotence is a function : H2 q (K(7, n); Fr )

H2,q+2(K (7, n

± I); Fp )

that is dual to the mapping sending x in H2q (K(7r, n);IFp ) to —/iPq (Tx) in B- 2pq+ 2 (K (7 , n ± 1); Fr). The transpotence is the extra piece of structure needed to design constructions inductively and so compute H.(K (7, n); IF,) for all n. Returning to the action of the Steenrod algebra, we observe that Ar acts on E2* ' 0 and E20 ' * as isomorphic to H* (B; ) and H* (F; Fr ), respectively. A natural question, posed by [Massey551, is whether there is an Ar -action on all of for all r, that converges to the action of Ar on H* (E; Fr). A solution to this problem was given independently by [Vdzquez571 and [Araki571 who proved the following result.

6.2. The transgression

195

Theorem 6.15. On the mod p cohomology spectral sequence associated to a E —> B, there are operations

fibration F

for p, odd

F Ps E,a„ ,b

Ea,b+24-1)

B ps : Ea,b

E,c!:±(28—b)(p-1),pb ,

for p = 2

FSq i : 1. BSq i : 4,1)

1 < r < 00, 2 < r < 00

Ea,b+i,

1 < r < Do

4,2b

2 < r < Oc

that converge to the action of A r on H* (E;Y r ), commute with the diff erentials in the spectral sequence, satisfy analogues of Cartan 's formula and the Adem relations and reduce to the A r-action on H* (B; 7p) and H*(F;Tp ) when r = 2 and a = 0 or b = 0 (that is, on E2" and E20*). These operations have been developed further by Kristensen621. [Kuo651 has given a general approach to the construction of families of operations on spectral sequences. [Singer731 has constructed Steenrod operations on certain first quadrant spectral sequences that include the Leray-Serre spectral sequence as an example. On H" (K(7r, n); 7p ); the theorems of Carton and Serre We can now turn to the major goal of this section—the computation of H* (K (7, n); 7p) for a fixed prime p and 7 a finitely generated abelian group. By the fundamental theorem for abelian groups and the fact that K(7,e72, n) K (71,n) x K (72, n), the task reduces to the consideration of the cases 7V = Z, EfezFE and Efekz17 for some prime L. We reduce the task further to the case = p. Lemma 6.16. If 7V is a finitely generated abelian group, n > 1, and k is afield, then if.(K (7,n); k) = {0} if and only if 7r k = {0 } . PROOF: If f-/-* (K(7, n); = {0}, then we have Hn (K(7, n); = {0 } . By the Hurewicz theorem and the Universal Coefficient theorem, IIn (K (7r, n); k) k and so k = {0}. If 77 k = {0 } , by the fundamental theorem for finitely generated abelian groups, we can fix it = E/ME for an integer m relatively prime to the characteristic of the field k. The fibration K (7r, n — 1) —> PK (7, n) —> K(7v, n) and the Leray-Serre spectral sequence for homology with coefficients in the field k show that if the lemma holds for n — 1, then it holds for n. It suffices to prove the lemma for (K (17 I m27,, 1); k). —Xm K (, 2)—>K (E, 2) where the Consider the libration K (El m22,7 , mapping, — x m, induces the 'times m' map on the second homotopy groups and hence on the second homology groups of K(, 2) = CP(co). Since E/m2,7 0 k = {0 } implies that m is a nonzero unit in k, the homomorphism

196

6. The Leray-Serre Spectral Sequence II

induced by — x m induces an isomorphism on H ( ;k) between the base and total spaces. This implies that the first nonzero vector space H (K ( mZ , 1); k) persists to E' and hence contributes to Hq (Z P (co); k) which has already been accounted for by the base space. This proves the lemma. We recall from Chapter 4 some facts about the Steenrod algebra. When pis an odd prime, a sequence of nonnegative integers I = (se , Si, Ei, . S m , Em ), with ri = 0 or 1 for all j, represents the operation St/ = /3 6° P91 36 i • • • P8 — 3'— in A. Such a sequence represents an element in a basis for A over 7, whenever the sequence is admissible, that is, si > psi±i + ei for m > i > 1. When p = 2, we write such a sequence as I = (S i , . . . S m ) to which we associate the operation St' = Sq" Sqs 2 • • • Se— E A2. A sequence is admissible mod 2 when si > 2si for in > j > 1. ,

Definition 6.17. For p an odd prime let I = (so, si, si, • • • , 87,2 , sm ) be an admissible sequence. The degree of the associated operation is denoted by and is given by I = 2(p — 1)(si + • • • + sm ) + Eo + • • • +Ern . lf x is a class in Hr (X; 71, ), then St i x is a class in Hr+111 (X; 7, ). The excess of I is given by e(I) = 2(si

+

—

p 5 2) + 2(82 —p5 3 ) +

2S m +EO8i — • • • —

+ 2(sm _ i — ps, n )

Em

= 25ip + 2E13 — IL

When p = 2, 1= (si, . . . ,Sm), I = si + • • • + sm and e(I) = 2si —

Before stating the main theorem, we state a rather technical lemma that describes how admissible sequences of a certain excess can be constructed from those of lower excess. The proof is an exercise in the definitions of admissibility and excess, with some careful bookkeeping, and is left to the reader. This lemma is crucial in the proof of the main result of this section. Lemma 6.18. Let p be an odd prime and define the admissible sequences = ,O,Pk,O,k) = (1 ,1A,Jkt-1 ). Suppose I = (so, Si, El, • • • , Sm, Sm) is an admissible sequence with e(I) < n for some n. (1) When n+1,11 is even andt > 0, let k = (1/2)(n+1/1). Write JoI = (J, I) for the concatenation of admissible sequences. Then .J, I is admissible and e(JkoI) = n. Furthermore, Jko I is admissible ande(4oI) = n+1. (2) If K is any admissible sequence with e(K) = n +1, then there is a t > 0 and a subsequence I of K with e(I) < n such that K = J 0 I and 2k = n+ 1 +1/1; or K = JZo I with 2k = n+ IL or K = J 0 o with / i = l2pt' +1 + land 212 = n+ 1+1 1- 1.

The analogous lemma for the prime 2 is left to the reader.

197

6.2. The transgression

Theorem 6.19 Kartan54]). Let p be an odd prime. (1) H* (K (ElpE,n); 71,) is a free, graded-commutative algebra on generators St', where in denotes the fundamental class in H"(K(ZIpZ,n); 7p) and I is an admissible sequence with e(I) < n or with e(I) = n and I = (1, s, 1'). (2) H*(K(E,n); 7p) is a free, graded-commutative algebra on generators St i i,„ where in denotes the fundamental class in H 2 (K(17 ,n); Fp) and I is an admissible sequence, = (so, s i ,s i , ,s,,E,), with 8, = 0 and e(I) < n, or e(I) = n and I = (1, s, I'). (3) For k > 1, H*(K(ElpkZ,n); 719) is a free, gradedcommutative algebra on generators Stile in where in G Hn(K(ZIp k E,n); 7p) and I = (80,81 , E l , . . . ,S m , E,) is admissible with e(I) < n or e(I) = n and = (1, s, I'), and if Em = 1, then Stki n = St(E° ' 81 'El— ' s—) (Okin), where 13k is the leh power Bockstein operation. Theorem 6.19 (ISerre531). (1) H*(K(Z/2Z, n);72 ) is a polynomial algebra on elements Sti in, where in E Hn(K(Z12E,n);72) is the fundamental class and I = (Si, 89, ... ,8,) is an admissible sequence with e(I) < n. (2) If n > 1, then H*(K(2,n);72) is a polynomial algebra on generators St' i n where in G Hn(K(Z,n);F2) is the fundamental class and I = (8 1 , s2 ,... ,s,) is an admissible sequence with e(l) < n and s, > 1. (3) For k > 1, H*(K(E12 k 7Z,n);72) is a polynomial algebra on elements StLi n where i n E Ha (K(7,12 k E, n); 1F2) is the fundamental class and I = (S i , s2, ,s,) is an admissible sequence with e(I) < n and if 5, =1, then StLi n = St( * 1— '.9-1) 13kin, where 13k is the eh power Bockstein homomorphism.

The kth power Bockstein operation is defined by examining the long exact sequence in cohomology that arises from the short exact sequence of coefficients, 0 —>

Ipk-F 1 Z ZIp k Z

0.

The connecting homomorphism carries the fundamental class,

in ,

to /30,,;

• • • —> Hn (K (Z p k 2Z, n); 1F1 pk Z)—> i3k H71 +1 (K (Z p k Z,n); 7,)

•••.

See Chapter 10 for other details of such operations. We present the proof of Theorem 6.19 in detail for p an odd prime and it E/pE. This case was first computed by [Cartan54] using acyclic constructions. Our proof follows the argument of [Serre531 for the case p = 2 as outlined by [Postnikov661 for odd primes. We proceed by induction. The key to the general inductive step is a theorem of [Bore1531 for which we need an odd primary notion that substitutes for the mod 2 simple system of generators (Definition 5.12). Definition 6.20. A graded-commutative algebra over IF, W*, is said to have a p-simple system of generators {x j G J} for J some totally ordered set, if

198

6. The Leray-Serre Spectral Sequence II

W* is generated as a vector space over Fp by the monomials 47'4' • • • x7: where j 1 < i2 < • • • < ik and, for 1 < j < k, 0 < mi < p — 1, if deg xi, is even; 0 < mi < 1, if deg xi, is odd.

A canonical example is IE Au], the polynomial algebra, with u of even degree. The uniqueness of the base p expansion of a natural number implies that {u, uP, uP2 , . . . } is a p-simple system of generators. This system of generators can be represented using Steenrod operations: Suppose deg u = 2k, then the sequence becomes {21, Pk 11, PkP (Pk 11), Pic132 (Pk P (Pk U)), 1. The reader can compare the next theorem with Example 1.1 and Theorem 3.27. Theorem 6.21. Suppose {E",d r} is a first-quadrant spectral sequence of algebras over Fp satisfying the following hypotheses: (1)

E2"

V* 0 IV* with E2* '0 V* and E2° ' * W* as algebras,

(2) E* is trivial,

(3) IV* has a p-simple system of transgressive generators {x i i E J}. Then V' is a free, graded-commutative algebra on generators {yi , z i , j E J} where yi = 7- (xi) and zi = r(yi ® 4 -1 ) when x j has even degree. Here T denotes the transgression, deg yi = 1 :h deg xi and deg zi = 2 + p deg xj.

PROOF (Riore153, §§16, 171): The proof follows the same outline as the proof of Theorem 3.27. We introduce two versions of elementary spectral sequences: For s odd, let E(s) denote the spectral sequence with E(s)"

Ip [y]

A(x)

where bideg x = (0, s), bideg y = (s + 1, 0), and T(x) = y. We can picture E(s) in this case as in the diagram: yx

y2 ®x

Y

Y2

y 3 0x

y 4 0x

Y

4

The spectral sequence E(s) satisfies the hypotheses and conclusion of Theorem 6.21 and furthermore, Fp [y] has a p-simple system of generators. For s even, let E(s) denote the spectral sequence with E(s)"

A(y) F[z]

FP[x 1/4xp _ 0)

where bideg = (0, s), bideg y = (s +1,0), bideg z = (2 + sp, 0), r(x) = y and T (y0x 2-1 )= z. We picture E(s) in this case as in the diagram on the next page. Once again, E(s) satisfies the hypotheses and conclusion of the theorem.

199

6.2. The transgression

z0xP -1 zy0xP -1

zox3 z0x2 zy0x2 zy0x zOx zy

Suppose the hypotheses of Theorem 6.21 hold and deg xi = si for the generators of W*. Form the spectral sequence, 0 E(si ) together with the iEJ

mapping of E2 -terms, E ( s i )02 ,*

E(2),*

iEJ

E(si )*2 '*

E2"

that induces an isomorphism

W. By Zeeman's comparison theorem (Theorem

we deduce that 0 E(s i ) *2 '° E2* ' 0 V* is an isomorphism of iFJ spaces. Examining E(si )*2 ' ° in each case, we find the expected free, gradedcommutative algebras. It remains to show that V* is free. The argument is similar to the one given in Example 1.K and so we leave it to the reader. 3.26)

7,-vector

To prove Theorem 6.19 we apply induction on the integer n and the Lerayspectral sequence for the fibration

Serre

K(ZipE,n) PK(ElpE,n +1) —> K(ZipZ,n +1).

When n = 1, K(E/pZ, 1) has the homotopy type of the lens space L" (p), which can be constructed as the direct limit of orbit spaces, s2k+1

s2k+3

/ Z/pZ

/ E/pZ " ' •

Here 82k+ 1 is taken to be the unit vectors in C k+1 and EME acts diagonally as the group of pth roots of unity in L. It is elementary to compute H* (L"(p);IF,) from the cell structure (see, for example, [Whitehead, GW78, p.921): * (K (ZIPZ, 1 ); 713) = A(ii)

FP

where 2,1 E H' (K(E/pE, 1); Tp) is the fundamental class. To begin the induction, we establish that H* (K (277p17 , 1); IF) satisfies the conditions of Theorem

6.

200

The Leray-Serre Spectral Sequence II

According to the definition of excess, e(I) = 0 can only be satisfied by the admissible sequence (0) giving the element i. When e (I) = 1 we have the (j1)pt sequences (1) and (0,pt,o,pt—i,... 0,p, 0, 1, 1), corresponding to for t > 0. Only (1) produces a generator, and the sequences 4 0 (1) determine a p-simple system of generators for H* (K (17 Ip17 ,1); Fr) . We apply Borel's theorem as the inductive step. Suppose that the algebra H*(K(71.4Z,n);7 p ) is given as described in the theorem. If I = (E o , 8 1 , E 1 , ,8m, E m ) is an admissible sequence with e(I) < n, then the generator St', transgresses to St iin+, by Corollary 6.9 and the fact that i n in±i • When St/ in is of even degree, it contributes the p-simpletransgeo system of generators {Sti 2 n , (Strin )P, (S't/ in )P2 , to W*. We can write this system as { St/ in , Pk St i tn , PkP(Pk St / in ), } 6.19.

where 2k = n+ I. These elements transgress to generators Stik °/ 2 7,4 1 where J,t, is described in Lemma 6.18. Thus H* (K (ElpE, n); Fp ) has a p-simple system of transgressive elements. The spectral sequence for the path-loop fibration over K(ZipZ,n + 1) provides the rest of the hypotheses and so H* (K (E pE, n 1); ) is a free, graded-commutative algebra on certain generators. It remains to determine that the generators are the ones described in the theorem. When deg(Strin ) is odd. Then St i tn+i generates a polynomial subalgebra of H* (K(2/pZ,n + 1);Ep ). When deg St rin is even, let 2k = n + {/- 1 and we get elements {St r in+1, st7n+i, ,s,t40/ 7n+1, in [pad (K p7, n ± 1), iFp, ) all of the form Sty' in+i where e(I' ) < n. The new generators take the form {T[St i in+i

(Stip-1], [pk (s tl in+i) )

(pk s ti

in)

j,

.}. ..

Here 2k = n + I with e(I) < n. The Kudo transgression theorem implies that T[St4c1 in ±i (Si4clin)P-1]

,(3pkp t stJkoI in+i

SO:

o in +1

From Lemma 6.19 we conclude e(Je+1 0 = n + 1. Since the converse part of Lemma 6.19 describes all admissible sequences of excess equal to n+ 1, we have proved that H* (K(/p,n+1);Fp ) is a free, graded-commutative algebra on generators StK in+i where K is an admissible sequence with e(K)
6.2. The transgression

201

of p, we can represent K (E p k E,1) as the lens space L"(p k ). By a careful accounting of the Bockstein homomorphism, it is elementary to prove that H* (L"(p k );Y p )

A(2, 1)

Fp[1301]

and the induction is off and running. Finally, we remark that the analogous argument works for the prime 2. 0 Having completed these calculations, let us put them to work. For the rest of this section, we fix p = 2. (The case of odd primes is similar but the bookkeeping is more complicated. See the paper of [Umeda59].) In this case all the algebras involved are free and strictly commutative and have simple systems of generators as in Definition 5.12. Recall from Example 1.F, the definition of the Poincaré series of a space, here given for coefficients in the field E9 2: P(X,t) =

i=0

(dimF2 H' (X ;7 2))ti

Let P(7, q, t) = P(K (7, q), t). Theorem 6.19 states H*(K(E72E, q);7 2)`-=' F2 [S/1- 2q I = (si , . . . , sm ), admissible, e(I) < 1

It follows that P(EI2E,q,t) = /, admissible,

1—

e(I)
In what follows, we make a closer analysis of these formal power series. To wit, consider an admissible sequence I = (Si, s2, ,Sm). Since si the sequence can be described by an associated one, (al, t2,... , a m ) defined by a m = sm and, for i < m, ai = si — 2si±i . Notice that the excess (how much si exceeds 2 s+ i) is equal to Ei cut . If we denote a typical factor of P(7,72Z, q,t) by 1/(1 — tin, then N = q + , and N—q= 1 =

si =

, i-1

ai (2 i - 1).

The last equation can be seen as follows; sm = am 8 m-1 = am-1 + 2s m = am _ i ± 2a m 8 m-2

= Ce m _ ±

=

2cem _

Llarrt,

and one continues by induction. Introduce ao = q — 1 — to write

us

2 =Nq= i=1 rn = ai 2 i - q +1. i=o

j=i 5

i=1

ai .

This allows

t=

ai 2i — (q — 1 — ao)

6.

202

The Leray-Serre Spectral Sequence II

Thus I is determined by a sequence (no, a i , for which n=

q + 1 11 = +

, am )

whose sum is g — land

cti 2 '

= 1+ 2° ± • • • ± 2 ° ± 2 1 ± • • • ± 2 1 ± • • • ± 27n ao

± • • • ± 27n arn

al

q-1 powers of 2 = 1 + 2 1"- + 2 162 + • • • + 2 hq-i

with 0 < hi < h2 < • • • <

.

This implies the following expression. Proposition 6.22. P(E/27„ g, t) = 0
1 - t1+2hi

+2hq-1

A similar development for P(Z,g,t) yields

fi

P(Z, q,t) =

1 1

t1+2hi

0
In the case of higher order torsion, a higher order Bockstein takes the place of the ordinary Bockstein, and it follows that P(ZI2 k Z, g, = P(z/n, g, t). We record some facts about these Poincaré series with regard to their analytic properties. For proofs, the interested reader should consult the proofs of ISerre531, lAhlfors66, p. 191] or the local analytic number-theorist. Fact 1. For 7F, a finitely generated abelian group, P(7, g, t) is a convergent series, on the open unit disk It G 1E; t < 1} with an evident essential singularity at t = 1.

In order to study the behavior of these functions near t = 1, Serre introduced the functions, defined for x large, = log2 P(7, q, 1 — 2 — x).

Fact 2. 0(E/2E, q, x)

xq

7 , and 0(E, g, x) if lim f (x) I g(x) = 1. q' x—> co 1

5 q-1

(g — 1)!

, where f(s) g(x)

Since P(Z/2Z, 1, t) = (K(/2, 1) (oo)), 6(Z 12Z,1, x) = x. 1— t The proof of Fact 2 follows by induction and a careful use of the product expansion in Proposition 6.22. From these facts about the Poincaré series of Eilenberg-Mac Lane spaces, we prove a remarkable theorem of ISerre531.

203

6.2. The transgression

Theorem 6.23. If X is a finite CW-complex that is connected and simplyconnected, and X is not contractible, then 71-i (X) contains a subgroup isomorphic to Z or ?Z/27Z for infinitely many i.

X,

Toward the proof, we introduce another construction associated to a space a tower of fibrations with X at the bottom.

K(72 (X), 1)

X(3)

K(71 (X), 0)

X(2)

X

At each successive stage, we "kill" the bottom dimensional homotopy group of the previous stage by attaching the appropriate series of higher dimensional cells. Definition 6.24. The upside-down Postnikov tower of a space X is a tower offibrations satisfying the properties: (1) X2) is the universal cover of X,

(2) the composite map, p: X (n) —> X induces an isomorphism, if j > n, 7ri (p): 7ri (X (n)) —> 7r i (X).

(3)

(X (n)) = {0} for j < n.

The homotopy long exact sequence for each fibration X (n) —> X (n — shows that the fibre is a K(7rn_i(X),n — 2). Such towers were introduced independently by [Cartan-Serre521 and by [Whitehead, GW521. The upsidedown Postnikov tower can be related to the usual Postnikov tower by observing that the space X(7 -0 can be constructed as the homotopy-theoretic fibre of the mapping X Pn X where Pn X is the nth space in the Postnikov tower for X. Up to homotopy we have the libration X(7 -0 —> X —> PX. PROOF OF THEOREM 6.23: Suppose N is the largest integer for which 7rN (X) contains a subgroup or z7I2Z. Consider the libration K (7rN

(X),

N — 1) —>

X (N ± 1) —> X (N).

If * (x + 2 ) = {0}, then the Zeeman comparison theorem implies H" (X (N);72) H* (K (7rN (X), N);72) . We prove the vanishing of

6.

204

The Leray-Serre Spectral Sequence II

B-*(X(N+1); 72): ByLemma6.16, ii* (K(7N±k(X),N+k-2);72) = {0}

for all k> 0 and so the mod 2 cohomology Leray-Serre spectral sequence for the fibration K(7N+k (X), N + k — 2) —> X (N + k) —> X (N + k — 1) collapses to yield H* (X (N + k): 72) H* (X (N + k —1);72). Stringing these isomorphisms together we have that H* (X (N +1); 72) H* (X (N + k);72) for all k > 0. Since X (N + k) can be made to be arbitrarily highly connected, 1-/*(XN + 1); 7 2 ) = {0 } . Recall, from Chapter 1, some of the properties of Poincaré series: (1) If {E'r", dr.} is a first quadrant spectral sequence, converging to H*, then P(E2", t) > P (H* , t). (Recall that P (A, t) > P(B,t) if the power series p(t) = P(A, t) — P(B ,t) has all nonnegative coefficients.) In E —> j) B is a fibration, then P(B, t) x P(F,t) > particular, if F P(E,t).

(2) If 7V = in ED 72, all finitely generated abelian groups, then P(7 , q, t) = P(71 , q, t) x P(72, q, t) . (Here x denotes the Cauchy product of power series.) When we apply these observations to the upside-down Postnikov tower, we have P (X (N),t) = P (7rN (X), N,t) < P(7N _ 1 (X), N — 2, t) x P (X (N —1) ,t) < P(7 N

(X), N — 2, t) x P(7riv _ 2 (X), N — 3, t) x P (X (N — 2), t)

< P (X ,t) >K

N-1 Hj=2 P(7rj (X), j — 1, t)

Because these power series have only positive coefficients, the inequality of power series gives an actual inequality on evaluation when t is in the interval [0, 1). Also, log2 is an order-preserving function. Thus, for x large, we have 0(7 N(X), N, x) < log2 (P(X, 1 — 2 —x )) +

N-1

j=2

0(

(X), j

—

1,x).

If we express a finitely generated abelian group qv as a direct sum 72-7_,EDZe•• • e z z/2k , ED • • • ED 2-,72k-z Er)

then, at the prime 2, P(in, q, t) = (II P (E, q, t)) x

stuff of odd torsion,

(fi 11 P(E/2Z, q, t)) r

and it follows that 0(7, q, x) = Es, o(F, q, x) + Er, o(F/2F, q, x). From this equation we have that 0(7, q, x) rsq 1 q!, or, if r = 0, then 0(7, q, — sacq -1 1 (q — 1)! where s is the number of factors of F appearing in the decomposition.

205

6.2. The transgression

Since the space X is a finite CW-complex, P(X, t) is a polynomial in t P(X,1 — 2 — x) = P(X ,1). Thus, log2 (P(X, 1 — 2 —x)) and so 1im a constant. Putting this together with the inequality bounding P (X N), t), we see that the left hand side (lhs) 0(7rN (X), N, x) converges like rNx N IN! or 8Nx N-1 I(N — 1)! as x grows unbounded, depending on the decomposition of 7rAT (X). On the right hand side (rhs) of the inequality, we find the relation log2 (P(X, 1 — 2 —x))

N-1

0(7ri (X), j — 1, x)

A

X

N-2

(N — 2)!

However, as w goes to infinity, the inequality must be violated at some a. since X

N

TN — N!

lhs < rhs

A

X

N-2

(N — 2)!

This contradiction to the existence of such an N establishes the theorem. Around 1950, little was known about homotopy groups. These groups were known to be denumerable for finite CW-complexes. For spheres, 7 (Sn ), 73 (S2 ), 77 (S4 ) and 715 (S8 ) had been computed and shown to be isomorphic to 7Z+ finite. Other cases were computed by [Freudenthal37] (7rn+i (Sa) 127) and [Whitehead, GW501 (7rn+2 (Sa ) 21 /2E). It was not known whether the groups were nontrivial after a certain range of dimensions (by analogy with homology groups). In his thesis [Serre51] proved that (X) is finitely generated when X is a simply-connected, finite CW-complex (Proposition 5.17). He also established the finiteness of 7ri (Sn ) for i n (and i 2n — 1 for n even) as in Example 5.K. With the further development of these methods (Theorem 6.23) [Serre53] proved the nontriviality of infinitely many homotopy groups of any noncontractible finite complex. To quote [Whitehead, GW831 from his history of homotopy theory, "It is no exaggeration to say that Serre's thesis revolutionized the subject" The analogous theorem for odd primes given by Theorem 6.23 was proved by similar means by [Umeda59]. The theorem for all primes has been extended by [Mc Gibbon-Nei sendorfer84] to remove the finiteness assumption and replace it with the conditions that X be simply-connected, II (X ; 1F,) {0} for some n > 2, and H (X;7) = {0} for n sufficiently large. They conclude further that there is p-torsion in infinitely many homotopy groups of X. Their proof relies on the study of [Miller, H84 1 of the space of mappings map(BE/ 2E, X) for X a finite complex. The upside-down Postnikov tower may be used to prove a generalization of the rational Hurewicz theorem (Theorem 5.18) to fields of arbitrary characteristic. Theorem 6.25 (the Hurewiez-Serre theorem). Suppose that k is afield and X a simply-connected space offinite type. If Hi(X; k) = {0} for 1 < i < n, then

6. The Leray-Serre Spectral Sequence II

206

We show by induction on j that, under the assumptions, Hi (X (j); k) = {0} for 0 < i < n and 0 < j < n. For j = 1, X(2) = X because X is simplyconnected. Furthermore, 71(X) k = {0} for trivial reasons. PROOF:

The key to the induction step is the Leray-Serre spectral sequence for the fibrations K (7ri (X), j — 1) —> X ± 1) —> X (P . We assume that Hi (X k) = {0} for 0 < i < n. The E2 -term for the fibration takes the form Ep2,q Hp (X (j); k) k Hq ( K(7ri (X), j 1); k). If 7ri (X) Hi _ 1 (K Ori (X), j 1); k) were nonzero, then {0} and since g o = {0} for 0 < i < n and j < n, there is no nonzero differential to hit and so Hi _ j_ (X (j + 1); k) {0}, which contradicts the fact that X(j ± 1) is j-connected. Thus 7ri (X) g k = {0} for j < n and, by Lemma 6.16, — 1); k) = {0}. This implies that H* (X ± 1); k) H* (X 0; k) and the induction is complete. When j = n — 1, we have shown that Hn (X ; k) (n); k) -

—

Ea

—

7„(X) k.

In the special case of X = S 271-1- we can be very explicit about the computation. The following result of [Serre511 extends our knowledge of the homotopy groups of spheres. Proposition 6.26. For n > 1, Hi (S2n -1 (2n); Fp ) = {0} for 0 < i < 2n + 2p — 3 and H292+2p_3 (S2n-1 (2n); 7p ) 7p.

PROOF: Since we know that K(Z, 2n

72_1(S 2n -1 ) E,

2)

—

we have the libration s2n i.

> S2" -1 (2n)

—

—

By Theorem 6.19, H* (K(, 2n — 2) 1F,) is a free graded-commutative algebra on generators St/ i2_2 for admissible sequences I = (60, Si, 61, ,S m , Ern) with Em = 0 and e(/) < 2n —2. The transgression takesn-2 to thegenerator u H271-1 (S2n -1 ; Fp) and the powers 4 map to :;,20i i 2n-2 for j < P. The class in least degree that persists e to27E:±a2 11d p determines the connectivity of H* (S2n-1 (2n); 7p ) is Pli2 ri_2 ( 1) (K(Z, 2n — 2); Fp ) . Since Pl corresponds to the admissible sequence (0, 1, 0) of excess 2, P l i2n _2 is found in H* (K (Z, 2n — 2), 7p ) for all n > 1. (s2n-1 ) for j > 2n —1 and so the By construction 7ri ( s2n-1 27.1)) Hurewicz-Serre theorem implies the following result.

= {0} for 2n

Corollary 6.27. 7r(S 21 ) 7 2n±2p-3 (S 217-1 ) 0

—

1 < i < 2n 2p

—

3 and

•

Analogous results for even-dimensional spheres can be obtained by similar arguments. The argument for Corollary 6.27 and the Freudenthal suspension provide the basis for an induction that motivates the Adams spectral sequence of Chapter 9.

6.3. On classifying spaces and characteristic classes

207

6.3 On classifying spaces and characteristic classes

Among the successful applications of homotopy theory, few are as spectacular as those found in the study of vector fields on manifolds. It was already known, in the "ancient" origins of the subject, that geometric problems of this sort could be tackled with topological tools. For example, the Poincaré index theorem (Poincaré18811) relates the index of a vector field to the Euler characteristic of the manifold. Certain problems on differentiable manifolds can be seen to involve, if not reduce to, the problem of the existence of linearly independent vector fi elds on a manifold. An example is the problem of embedding an m-dimensional manifold M in a Euclidean space RN . Such an embedding leads to a bundle of (N —m)-dimensional vector spaces vm over M of normal vectors to the tangent space. A necessary condition that the manifold be embedded in a dimension lower in RN-1 is a nonzero vector field of vectors in vm. Another problem that can be formulated this way is the problem of the existence of a division algebra structure on Rn (see [Adams60] and Chapter 9 for this idea). [Stiefe1361 in Switzerland [Whitney351, then at Harvard, introduced the idea of characteristic classes toward solving the problem of determining the number of linearly independent vector fields on a manifold in a given vector bundle. Following [Steenrod511, these characteristic classes arise as follows: Let (Rn ) denote the Grassmann manifold of k-dimensional linear subspaces in . Triangulate the manifold M and suppose there is a bundle of n-dimensional real vector spaces over M. Assign k linearly independent vectors at each vertex of the triangulation. This gives a mapping, MO —> C k (Ea), sending each vertex in the 0-skeleton of M to the k-dimensional subspace spanned by the k vectors in an identification of the vector space at a point with R. The problem is to extend this assignment from the 0-skeleton to the 1-simplices, and then to M( 1 ), the 1-skeleton of M, and so on, proceeding a dimension at a time. The obstruction to extending the assignment at dimension (i — 1) to an assignment on the i-skeleton is a cocycle in Ci (M i_ (G k (RI))) , the cochains on M with coefficients in the system of local coefficients, 72-1(Gk (the vector space at each point in M)).

By reducing mod 2, these obstructions determine classes wi G Hi (M; 72) now known as the Stiefel-Whitney classes of the bundle under consideration. Analogous constructions given by [Chem481 for bundles of complex vector spaces and by Wontryagin471 are important tools in the study of manifolds. Characteristic classes are useful in the classification of vector bundle structures on manifolds. One can reverse the process by building vector bundles on manifolds using the properties of specially constructed spaces called classifying spaces. The algebraic topology of classifying spaces is the subject to which we next turn the Leray-Serre spectral sequence. We first introduce some of

6.

208

The Leray-Serre Spectral Sequence II

the fundamental notions of vector bundles with structure and sketch the construction and properties of classifying spaces. We then study the cohomology of classifying spaces with the aid of the Leray-Serre spectral sequence. These computations were first done by [Bore153] in his celebrated thesis. Finally, we reconsider characteristic classes, having computed the cohomology of the classifying spaces and apply those results to some geometric problems. The details of the theory of characteristic classes would easily double the length of this book and take us, not unhappily, far afield of our central concern. There are many good expositions of these topics, for example, the papers of Riore153, 55 1 , and the books of Wilnor-Stasheff741, [Dupont781, [Husemoller661, and [Bott-Tu821. Our goal is to demonstrate to the reader the crucial role played by the Leray-Serre spectral sequence in this development. On vector bundles and classifying spaces

We begin with the observation that the tangent bundle to a manifold is more than a fibration—it carries extra structure. Definition 6.28. A fibration F group G if the following hold

E>.13. is a fibre bundle with structure

(1) G is a topological group and there is an action of G on F, the fibre, tt: G x F F, satisfying, for all gi, 92 E G and all x E F, ft(gi, 12(92, x)) = ,a(g1g2,x) and ft(e, x) = x. Furthermore, the action is effective, that is, if 1.1(g,x) = x for all x in F, then g = e. (2) There is a family, {Ui j E J} of open sets of B that cover B and, for each j in J, a homeomorphism, : Ui x F —> p -1 (U1), satisfying the equation pcp (x, y) = x. (3) For each pair, i, j in J, and x in Ui n Ui, the mapping g 1 (x) defined by o (,o1 (x,

gii(x) = (pi(x,

—

): F

p-1 ({x})

F

is continuous in x and given by left multiplication by an element of G, that is, the map gii determines a continuous function Ui n U1 —> G and gii (x)(Y) =

(x), Y).

The principal example is the tangent bundle to a manifold. There the fibre is IP:n for M. an m-dimensional manifold, and the structure group may be taken to be Gl(m, R), the group of invertible (m x m)-matrices. If we consider only the unit tangent vectors at each point in M, we have a fibre bundle with S' as fibre and structure group 0(m), the group of invertible, real (m x m)-matrices A with At A = I. If the manifold M is orientable, then the structure group for the associated sphere bundle may be taken as SO (m). In order to classify the fibre bundles with given fibre F and structure group G over a fixed space B, we introduce a notion of equivalence of such bundles.

6.3. On

classifying spaces and characteristic classes

209

Definition 6.29. A morphism of fibre bundles,

(f ,J): (G, F, E,p, B) —> (G, F' , , p' , B') is a pair of maps, f: B —> B' and f: E —> E' such that the following diagram commutes,

)

B

B',

and J F : F —> F' over each point is a G-equivariant map, that is, for all g E G and x E F, f(p,( g ,x)) = p' (g , (x)). For B = B', we say that a morphism (1, f) - is an equivalence of bundles i f F —> , over each point, is a

homeomorphism.

A sufficiently differentiable mapping of manifolds, f: M —> N, provides a bundle morphism of the tangent bundles and a diffeomorphism of a manifold to itself provides an example of an equivalence of the tangent bundle with itself. The first reduction of the problem of classifying all fibre bundles over a space B with a given fibre and structure group is to focus on bundles of a certain type. Definition 6.30. A fibre bundle is said to be a principal bundle if it has structure group G and fibre G with the action of G on the fibre given by left multiplication.

Let = (G, F, E,p, B) denote a fibre bundle with structure group G. We associate a principal bundle to this bundle by taking the data vi : U1 x F —> p' (LTj ) and gii (x): F —> F and replacing F with G everywhere. Property (3) of Definition 6.28 guarantees that another fibre bundle results. We call the new bundle constructed in this manner the associated principal bundle to and denote it by Print) = (G, G, E, p, B). Suppose F is a G-space with an effective action ,u:GxF—> F. Suppose further that I/ = (G, G, E,p, B) is a principal bundle. We construct a fibre bundle over B with fibre F, structure group G, and tt the given action as follows: Let {Ui } be the open cover of B given in the principal bundle structure on E. Let Z =JJ U xGxF denote the disjoint union of the sets U; x G x F. Define an equivalence relation on Z for which an element of Ui x GxF is denoted by (x, g, f) and we require, for x E n ui , ( x, g, f) (x, j, g ii (x)g, f) and, for all h G G, (x, j, g, (x. j, gh,h f). We denote by E X GF the quotient space of Z by this relation together with the projection 23( [x, i, g, f ]) = x. The reader can check that 77 [1] = (G, F, E x G F,p, B) is a fibre bundle (see [Husemoller66, §5.3.21). This procedure is seen to be effective for our purposes in the next proposition. For a proof the reader can consult the classic books of [S teenrod51] or [Husemoller661.

210

6.

The Leray-Serre Spectral Sequence II

Proposition 6.31. Let = (G, F, E,p, B) denote a fibre bundle with fibre F and structure group G. Let n = (G, G, E' ,p' , B) denote a principal bundle over B with structure group G. Then is equivalent as a bundle to Prin() [F] and n is equivalent as a bundle to Prin(n [1]).

The proposition reduces the classification problem to determining principal G-bundles over B. If we view G as a group of transformations on the fibre G, then the motivating example of classification is the theory of covering spaces and the fundamental group; any covering space over a space can be constructed from the universal covering space. By analogy we seek a universal principal G-bundle over a space. The following proposition allows us to identify the defining property of such a universal principal bundle. It is also makes use of another ingredient in the construction of fibrations—the pullback of a fibration over a continuous mapping (4.3). Proposition 6.32. Suppose = (G, G, E, p, B) is a principal G-bundle over B, a Hausdorff paracompact space. Suppose f, g: X —> B are homotopic maps, then = (G, G, f* E, f*p, X) and ,g* - = (G,G,g*E,g-p, X), the induced principal bundles over X, are equivalent.

SKETCH OF PROOF: Consider the diagram with H a homotopy from f to g. f* f*

Xx

This diagram is completed by the homotopy lifting property for fibrations and so this gives the data for a bundle homotopy from f* E to g* E. One can argue from the unique lifting of paths that the bundle homotopy provides an isomorphism of bundles. For all of the details, see 1Husemoller66, §4.91. Definition 6.33. A principal G-bundle n = (G, G, E, p, B) is n-universal if (E) = {0} for i < n. If 7, (E) = {0} for all i, then we say that n is a universal bundle and we denote E by EG, B by BG. The space BG is called the classifying space for G.

To establish the "universality" of these bundles one supposes given a principal G-bundle, (G,G,E,p,B). Over each finite skeleton of B, a mapping can be constructed, B (k) BG, that pulls back to the given bundle. The obstructions to extending the mapping to the next skeleton lie in the cohomology of B with coefficients in the homotopy of EG and so they vanish. If we restrict B to be a CW-complex of dimension n, then a principal G-bundle over B is the pullback of an n-universal bundle over a continuous mapping. When we combine these ideas with Proposition 6.31, we have proved the following result.

6.3. On classifying spaces and characteristic classes

211

Theorem 6.34. If x(n) = (G, C, E(n, G), p, B (n, G)) is an n-universal bundle and B a CW-complex of dimension < n, then the set {equivalence classes of principal G-bundles over B} is in one-to-one correspondence with the set of homotopy classes of maps [B, B(n, G)]. Corollary 6.35. If x = (G, C. EG,p, BC) is a universal bundle for G and B of the homotopy type of a CW-complex, then the set {equivalence classes of principal G-bundles over B} is in one-to-one correspondence with [B, BC].

Some examples are in order: (a) Suppose G — ZI2Z. Then, for each n, we have the bundle Z/2a, —)

S —) RP(n)

where RP(n) denotes the real projective n-space. Thus RP (n) is a version of B(n — 1, Z/22,S). The series of equatorial inclusions gives the system Sn

- 571+1

RP(n)

> RP(n +1)

S"

>

RP(œ).

The direct limit is the 2/2Z-bundle Z/2Z —) S —> RP(oc) with S" weakly contractible, a universal bundle for E/2Z. Thus RP(œ) is a classifying space for (b)Recall from Chapter 5 the Stiefel manifolds of orthonormal k-frames in Rn , Vk (Ir ). Consider the mapping, Vk(R n ) —> Ck(Rr) which sends a k-frame in Rn to the k-dimensional linear subspace it spans. This determines a fibre bundle with 0(k) as structure group. A careful argument with the integral homology Leray-Serre spectral sequence shows that Jri (Vk, (11n )) = {0} for k n and
BG =

Un

),

x

where Gn isGxGx•••xG(n times) and — denotes the relations (to ,

, tn,gi, • • •

,g.)

(• • • ,ti ,... gigi+i, • • • ), (• • •

+ ti, .. • rgi, • • • ),

if t

=

if gi = e.

212

6.

The Leray-Serre Spectral Sequence II

The interested reader can refer to the cited papers for details. Here are some pertinent facts about this construction. (1)This definition can be extended to define classifying spaces for associative H-spaces. The role of associativity and applications of these more general classifying spaces can be found in 11Stasheff63, 63'], [May75] and [MadsenMilgram791. (2)The construction of [Milgram671 is functorial on the category of topological groups. An example of this functoriality at work is the case when one classifies the fibration associated to a closed subgroup H G.

The following diagram commutes: H

H

G

1

1

1

G

) EH

> EG

1 ) BH Bi > BG

Gill

BH is the classifying map. From functoriality where the mapping GIH and the long exact sequence of homotopy groups, we obtain relations among these spaces. First, the acyclicity of EC implies that 7ri (G) or more descriptively, that G has the same homotopy type as QBG. Indeed, if G is a simply-connected CW-complex, the Whitehead theorem implies that G ABC. Also observe that the classifying map GIH BH acts like the inclusion of the fibre of Bi: BH —> BC. Thus, for homotopy-theoretic purposes, we have the fibration inc

GIH BH

Bi

BC.

This plays an important role in Chapter 8 where we compute H*(GIH;k). (3)Finally, suppose G is a discrete group. Then 7r0 (G) = G and 7ri (G) = {0} 1. It follows that for i > O. Thus ri (BC) G and 7ri (BG) = {0} for i BC K(G,1) and EC is its universal covering space. Thus the theory of classifying spaces is a generalization of the theory of covering spaces. On the cohomology of classifying spaces

With Corollary 6.28, we have reduced the classification of principal Gbundles over a space B to the determination of [B, BC], the set of homotopy classes of maps from B to the classifying space BG. A crude way to study

6.3. On classifying spaces and characteristic classes

213

[B, BC] is to consider the image of elements in this set under the cohomology functor, that is, [f] : B —> BC is sent to

[f]

G [B , BG]

f* :

(BG; R) —>

(B; R).

We call the image, f* (H* (BG; R)) C H* (B; R), the characteristic ring of the bundle whose classifying map is f; nonisomorphic characteristic rings imply nonequivalent bundles. To make use of the characteristic ring for classification, we need then to compute H* (BC; R) for various rings R. We would like to apply the Leray-Serre spectral sequence to the fibration G EC BC. If we can work backward from knowledge of (G; R) and the fact that H* (EC; R) R, then we can compute H* (BC; R). This situation recalls the computation of H* (K (7r, n); where we knew the cohomology of the fibre as the inductive hypothesis and proceeded to compute the cohomology of the base space. The structure of H* (G; R) is based on the fact that cohomology of an associative H-space of finite type is a graded-commutative Hopf algebra. The algebraic structure of such Hopf algebras was first studied by 11-lopf411 for k, a field of characteristic zero and, for other fields, by Wore1531. Theorem 6.36 (the Hopf-Borel theorem). Let k denote afield of characteristic p where p may be zero or a prime. A connected Hopf algebra H over k is said to be monogenic if H is generated as an algebra by 1 and one homogeneou s x of degree > 0. If H is a mono genic Hopf algebra, then elmnt

2 and degree x is odd, then H A(x), (2) i f p 2 and degree x is even, then H —= k[x]/x s where s is a power of p or is infinite (H k[x]), (3) i f p = 2, then H k[x]/x s where s is a power of 2 or is infinite. (1) if p

If k contains the pth root of each of its elements, k is said to be perfect. A graded-commutative Hopf algebra H, of finite type over a perfect field k is isomorphic as an algebra to a tensor product of mono genic Hopf algebras.

The algebra generators in the isomorphism given in the theorem constitute a simple or p-simple system of generators for H. Thus all Hopf algebras of the type described have p-simple systems of generators. PROOF: We prove the first part of the theorem for the prime p and the field 7,, which is perfect. The reader can provide the characteristic zero case for himself or herself from the argument here. Chapter 10 also contains the result. If x has odd degree in H, a monogenic graded-commutative Hopf algebra, then x • x = —x • x, and if p is odd, then H A(x). If H has finite dimension and the generator x has even degree (or p = 2), then, for some h > 0, X h = 0 while x h-1 0. Such a positive integer h is called the height of the element

6.

214

The Leray-Serre Spectral Sequence II

x. Since x is also the element of least nonzero degree in H, x is primitive, that is, A(s) = 1 x + x 1. It follows that 0 = A(x h ) = (A(x)) h = (1

ex+x h

h

i=0 (i)

Since

(1f )

i)h

e

h—i

is congruent to zero mod p for all 0 < i < h if and only if h = p8

i

for some power s (Lucas's Lemma), the stated form for the monogenic Hopf algebra holds. When H is infinite-dimensional, no power of the generator

vanishes and H '' k[x]. To prove the second half of the theorem, we order the algebra generators of H by degree: deg xi < deg x2 < deg x3 < • • • . Let An denote the subalgebra of H generated by the elements x i , . .. , x n and 1. The degreewise ordering of the generators implies that the coproduct A is closed on An and so An is a sub-Hopf algebra of H. Likewise, An _ i is a sub-Hopf algebra of A. We write Bn = An fiAn _ i for An /(An+_, • An ) where Ant, is the degree positive part of An_ i (An _i = Ant, ED k) and Ant, . A, is the ideal of An of products from An+_, and A. This quotient is also a Hopf algebraconsitg and monogenic with generator . We admit for the moment that we can choose x n G An with x n i— ,,, by the quotient An, —> Bn and, more delicately, that the height of x n is the same as the height of fl . Let n: Bn —> An denote the algebra monomorphism n(A,) = x n . Consider the composite: hie OTI A1 ® B

A F4 A A , An 0 An An An

Applied to an element a (5:) (1(0pr) 0 0 p,(a

10pr

e An —> An Bn •

we get

= (1 Opr)(A(a) • A(x)) = a®

yli f

where the elements yi' and g' are determined by A on x n and a. In particular, equals —a ® and the composite is a monomorphism. Since An _ 1 together with xn = 7/( n ) generate all of An , the first part of the corn--

no yi'

incatl> tt posite, An_ 1 0 Bn An 0 An —> An is an epimorphism, and hence an isomorphism. This is the inductive step. Writing Ai = Bi, the induction can start and the theorem is proved. It remains to show that we can choose xn E An of the same height as E B. Let's assume that char k = p and either p = 2 or deg x is even. The case of interest is Bn k[] /r ) for some s > O. The problem is that there could be some nontrivial relation—it could happen that 4: E An+_, An

6.3. On classifying spaces and characteristic classes

215

and 4: 0 for all choices of xn that map to We show that we can choose a representative for x such that 4: = O. Consider the Frobenius homomorphism (: H —> H given by ((a) = ar. This mapping preserves the coproduct structure, the algebra structure when H is graded-commutative, and takes primitives to primitives. Let Z = (8 (2-1,„_ 1 ) and form the diagram of Hopf algebras (we have used the property that if CcBcA is a sequence of normal inclusions of Hopf algebras, then (AlIC)11(BlIC)'='

An _ //Z

A n il Z

B„

Since 7T is onto, there is a y E A n fiZ with 74y) = x n . We show that yP s = O. Consider ,A,(y) = yel-Floy+E (gdj. Since 74a3' ) = 0 = 7r(q), the Therefore ,A,(yP s ) = yP' 1+ 1 yP s , that is, elements a, a3i! lie in An _ i yi's is primitive. However, there are no primitives in Bn of degree p8 deg and there are no primitives in An _ //Z in that dimension. The relations between the terms of the bottom row imply the exactness of the sequence of primitives: 0 —> Prim (An_ //Z)

Prim(An //Z)

P rim (Bn )

Thus yl? = O. If we let w be an element in An with Qn (w) = y, then qn (wPs ) = yP s = 0 and so wPsE Z. Since Z = (8 (A_ 1 ), there is a class y E An _l with wPs = vP . But then (w v)Ps = 0 and qn (w — y) = y. We take w — y for our representative of xn and it has the same height as S

—

We apply this theorem to H8 (G; k), which has the structure of a connected Hopf algebra over k when G is a topological group (actually, as in §4.4, it suffices for G to be an H-space with homotopy associative product). If G has the homotopy type of a finite CW-complex, then H* (G; k), for k perfect, has a decomposition, as an algebra, H* (G; k) A(x i ,

, xr)

k[yi]Ry l

k[ys ]Rghs

with all hi
6.

216

The Leray-Serre Spectral Sequence II

Proposition 6.37. If G is a topological group of the homotopy type of a finite CW-complex, k a perfect field of characteristic p, and H* (G) has no torsion at the prime p, then H*(G; k) A(x4,. ,x,) as an algebra over k with each

xi of odd degree.

In this case, Theorem 3.27 applies to compute H* (B G; k). Theorem 6.38. If G is a topological group of the homotopy type of a finite CW-complex, k a perfect ,field of characteristic p, and H*(G) has no torsion at the prime p, H* (BC; k)"=' k[yl , . . . , yr] where Y to x, under the transgression. i

The computations in Chapter 5 provide data for this theorem. In Example we computed H' (U(n); R) A(x 4 , x3, . . . , x2,4 _ 4 ) for any commutative ring of coefficients. For a perfect field k, then, Theorem 6.39 implies that H* (BU(n); k) k[c2,c4, . ,c2„], where the polynomial generators ci have degree ci = i and each ci corresponds to x,_1 under the transgression. By the Universal Coefficient Theorem, we lift the generators to integer coefficients and so H* (BU(n)) 7,[c 2 , c4 , . . , c2,„]. The inclusions U(n) c U(n + 1) C • • • C U determine the infinite unitary group. Computing the inverse limit of the homomorphisms induced by the inclusions we find no lim i -term and so 5.F,

H (BU) Z[c2, c4, c6, • • •]• For the group SO (n) as in

Example 5.H (step 3), the Hopf-Borel theorem (and direct calculation) implies that H*(S0(n); F2) has a simple system of generators {x l , . . . , x n _ 1 }. By Theorem 6.27, we obtain H* (BSO(n); 7 2) [w2, , w„]. Taking the limit of inclusions S 0 (2) c S 0 (3) c • • • c SO, we get H* (B SO; 72) F2 [W21 W31 W41 • • • • To compute H* (B0(n); 72) recall that the determinant homomorphism together with the inclusion, SO (n) c 0(n), gives a sequence of maps det

SO(n) 0(n) —> 2421 —> BSO(n)

Bi

13det

BO(n)

BE/2z7,

any three of which form a fibration. Thus BSO(n) is a universal covering space for B 0 (n) and H1 (BO (n); F2) F2 with generator wi. If we apply the Cartan-Leray spectral sequence for the universal covering BS 0(n) —> BO(n), the E2-term is given by Ep2, HP (Z/227„; Hq (BSO(n); 72)). Since Z/ 2Z acts trivially on H- (BSO(n);72) and H* (z7,12E,7 2 ) H* (RP (cx));72) we can rewrite the E2-term as 4 * = 7 2 [Wl] ®72 F2 [W21 • • •

W TI] .

By studying auxiliary quotient spaces related to this covering (using Stiefel manifolds), we find that the spectral sequence collapses and so H* (B0(n); F2) 7 2 [W11 • • • 11,rd • Furthermore, H* (BO;F2) F2 [W11 W21 • • • ]•

6.3. On classifying spaces and characteristic classes

217

BU(n), or In these cases, we observe that a classifying map, f: B f: B —> B 0 (n) has its characteristic ring in H* (B) and H* (B: 72 ), generated as an algebra by the classes f * (c2) , • • • , f*(c2n) or f*(wi), • • • , f*(wri)• In

the next section, we relate these generators to the characteristic classes. Applications

We turn our attention, once again, to vector bundles and the problem of computing the characteristic classes associated to a bundle. Using the vector space structure on each fibre, we define the following operation on vector bundles: B) and 17 = (R71 , E' , B) are Definition 6.39. Suppose = (R n , two vector bundles over a space B. The Whitney sum of and 77, c e = (Rn ±m ,E E' ,p e p', B) is the vector bundle given by the pullback diagram: Rn±n1

x

Ped

B

(nag

tPX29'

BxB

The definition allows us to take the fibres of andq over a point in B and form their direct sum over that point. The pullback operation provides the appropriate topological glue. Stiefel-Whitney classes for a vector bundle wi ( ) in Hi ( Xi; 72), can be described constructively in the obstruction theory framework and they can be constructed via the action of the Steenrod algebra in conjunction with the Thom isomorphism as proved by [Thom521 based on the work of [W11501. We describe these classes uniquely through a set of axioms first suggested by [Hirzebruch661. Axioms for Stiefel-Whitney classes. Let = over a base space M.

, E, p, M) be a vector bundle

I. There are classes w() Hi (1;79) and wo() = 1, w = 0 for j > n. II. If f : B 1 —> B2 is a continuous mapping that is covered by a bundle map 6 that is a linear isomorphism on the fibres, then wi (6) = (f, f): 6 f*(tv,(6)).

HI. For two vector bundles, and 77, the Whitney sum formula holds;

-0k(e ED 77) =

1

IV. Let

-4 =

(R, E, p, RPM) be the canonical line bundle over RP(1), then

wi(q) O.

6.

218

The Leray-Serre Spectral Sequence II

The proof that these axioms uniquely determine the Stiefel-Whitney classes can be found in the book of [Milnor-Stasheff74, p. 86 1 . We establish the existence of Stiefel-Whitney classes by constructing them from the cohomology of the appropriate classifying space. To wit, let = (Rn , E, p, M) be a vector bundle over M and suppose is classified by a map : M —> BO(n), that is, following Theorem 6.34, is equivalent to fx(n)[R n ] where x(n): E0 (n) —> BO(n) denotes the universal principal 0(n)-bundle and fx(n) is the pullback over A. We define classes in H- (M; 7 2 ) for by the formula wi(

)

=

where wi G H (B0(n); 72) is one of the transgressive generators identified in Theorem 6.38. Theorem 6.40. The cohomology classes defined by the classifying map satisfy the axioms for the Stiefel-Whitney classes.

SKETCH OF A PROOF: We provide a series of observations (with references) as a complete proof would take us too far afield. First notice that Axiom I holds trivially from our definition and the cohomology of BO (n). Axiom II, the naturality of the Stiefel-Whitney classes, follows from the universal properties of the pullback operation and the definition of a bundle map to replace the bundle i with p6. Then the classifying map for 6, &: B1 —> BO, is homotopic to fc., o f. E( 1)

E(6)

EO

B2

) B0<

E(6)

The relations between the Stiefel-Whitney classes follow. Axiom III, the Whitney sum formula, hangs on an extra piece of structure. Consider the mapping (p,„ : 0(n) x 0(m) —> 0(n ± m) defined on matrices A E 0(n) and B G 0(m) by n,m(A,

=

(A 0 B

*

This group homomorphism induces a mapping, Bp„,,, : BO(n) x BO (m) BO(n m), whose key property is recorded in the following lemma.

6.3. On classifying spaces and characteristic classes

219

Lemma 6.41. The homornorphism, (B v„,,,„)* : H* (B0(n + m); F2) > H* (B0(n); 72 ) g H* (B0(m); I2),

satisfies (B cp,

(wi) =

Er+8_,

tor W.

A proof of this lemma may be found in [Bore153', §6]. The crucial feature of Borel's proof is the identification of an important abelian subgroup of 0(n), Q(n), of diagonal matrices with entries ±1. The subgroup Q(n) is isomorphic to the product of n copies of Eln and we can examine the induced mapping B inc : BQ (n) —> BO(n). Since BQ(n) is the product of n copies of BE/27-z, = RP(oc), we know its mod 2 cohomology: H* (BQ (n); F2) 72[yi, , ya ] where degy = 1 for all i. The representation theory of Lie groups may be pressed into service to identify the image of (B inc)* : H* (B0(n);F2) —> H* (BQ(n);72)

, y, } , Sym(y i , as the symmetric functions in {yi , , yn ). By a counting argument, (B inc)* is an isomorphism of H* (B0(n);72) onto the algebra Sym(yi , , yri ). From this rather deep structural result, the lemma follows. The Whitney sum formula can now be proved by examining the pullback diagram that describes the associated principal bundle for ED y: E(Prin(

n))

diag

EO(n m)

f'0(.(n)x x(m) E 0 (n) x EO (m)

M x M fxf 7 BO(n) x

+ M)

Finally, we establish Axiom IV for the canonical line bundle over RP(1) by considering the geometry of the bundle carefully. The canonical line bundle over RP(n) can be described by the total space, E(y) = ({± } , tY), where

E Sn and t E

R};

the first projection provides the bundle map. For n = 1, this description identifies E(-y1) as the open MObius band. It is an elementary fact that E(71) is not homotopy equivalent to RP( 1) x R, the total space of the trivial bundle. On the classifying space level, we can identify wi (-y1 ) with [f.).1] in [RP(1), BO(1)]. Since 0(1)"=' Z/2E, we compute: [RP(1), BO(1)] = [RP(1), BZ/2Z] = [RP(1), K(27,127,,1)] = (RP(1); E 2 ). Finally, RP (1) is homeomorphic to S', and so tuj(-y1) is either the generator of H' or zero. If wi (71 ) = 0, then f rl would be null-homotopic and -y1 would be equivalent to the trivial bundle. Since this would imply the equivalence of the total spaces, the observation shows that wj (-y1) = O. 111

6.

220

The Leray-Serre Spectral Sequence II

Similar axioms were given by [Hirzebnich661 for the Chern classes of a complex vector bundle over a space M. In the original paper of [Chem48], these classes are constructed using de Rham cohomology and specific differential forms, which have relevance to developments in mathematical physics (see also [Milnor-Stasheff74, appendix C], [Dupont781, or [Bott851). In our formulation, the classifying space for n-dimensional complex bundles is BU(n) and so we can take the construction of the Stiefel-Whitney classes and carry it over to define, for a complex bundle, If ( M)• Notice that the Chern classes lie in the integral cohomology of M (a fact related to the quantization of magnetic charge; see the papers of [Dirac351, [Atiyah79], and, for an elementary exposition of these topics, [McCleary921). For a complex vector bundle, = (Ca , E, p, M) we have its Chem classes c2() H2i (M). These classes reduce mod 2 to determine classes H 2i (M; 72). Furthermore, by identifying Cn with R2', we obtain StiefelWhitney classes for E Hi (M;72). How do these reduced Chem classes, ( ) , relate to the Stiefel-Whitney classes? C2i

=

(e2i) E

Theorem 6.42. For a complex vector bundle, = (C a , E,p, M), the StiefelWhitney classes of the associated 2n-dimensional real bundle satisfy the relations w2, ±1() = 0 and w2,() = 2,,(), for 0 <j
U(m - 1)

) 0(2m — 1)

U(m)

0(2m)

U(m)/U(m — 1) = 8 2' 1 = 0(2m)/0(2m — 1).

This gives a mapping of fibrations s2m-1

BU(m — 1) B(C)I

s2m-1

B0(2m —1)

B ine

BU(m) B(C)1

B0(2m).

Other constructions of the spectral sequence

221

In the Leray-Serre spectral sequence for these librations the class G E2u ' 27n 1 corresponding to the generator of H 2 m–1 (S2'71-1 ;172) transgresses to w2,, E H 2ni (B0(2m); F2) and C 2m E H2m (BU(m); F2), respectively. If we identify BU(1) with BSO (2), then we can prove inductively that B(c)

H* (B0(2m);7 2)

> H* (BU(m);7 2)

takes /02i+ to zero and u) 2i to !f),i by the natural ity of the transgression modulo its indeterminacy. The theorem follows by observing that the classifying map for as a 2n-dimensional real vector bundle, factors through BU(n) —> BO (2n). An immediate geometric corollary of these relations is the following result. Corollary 6.43. Suppose M is a real 2n-dimensional manifold with tangent bundle TM• If M can be obtained from an n-dimensional complex manifold by the usual identifications, then w2j+1(TM) = 0 and w2i(TM) is the reduction of an integral class for 0 < j < n.

Thus the Stiefel-Whitney classes provide obstructions to the existence of a complex manifold structure on an even-dimensional real manifold. The satisfaction of these conditions is not sufficient to guarantee a complex structure (see the paper of [Mass ey611) but these classes provide a first set of obstructions. In this short discussion of classifying spaces and characteristic classes we have touched on very few topics in a rich area of study. The reader who is interested in these matters will find a wealth of applications and deeper developments in the literature. The role of spectral sequences in this work has been firmly established. 6.4 Other constructions of the spectral sequence

In Chapter 5, we constructed the Leray-Serre spectral sequence for simplicial singular homology and identified the E2 -term. In this section we discuss some different settings that lead to the construction of a spectral sequence for librations. We supply some of the details of an elegant construction due to [Dress671. The apparatus introduced there has other applications in topology; for example, it provided [Singer731 and [Turner981 with a useful framework in which to construct Steenrod operations in spectral sequences. The motivating examples of fibre spaces are the homogeneous spaces, H GIH, where G is a compact, connected, Lie group and H a closed subgroup. Investigations of the topological invariants of Lie groups had proceeded using analytic tools like de Rham cohomology and the associated Lie algebra cohomology ([Koszti1501, [Cartan511). The problem of relating the homology groups of the fibre, base and total space was undertaken by J. Leray in a series of boldly original papers that appeared from 1946 to 1950 ([Leray46,

222

6. The Leray-Serre Spectral Sequence II

50]). Leray based his investigations on a cohomology theory that generalized features of de Rham cohomology to more topological settings. Suppose f: E —> X is a surjective mapping where X is a compact polyhedron with a good covering, If; good is taken here to mean that any finite intersection, U 1 n U n • • • n LT,„, of subsets in the cover is contractible. Because f is surjective, this covering lifts to a cover of E, rill. As an instance of Leray's theory, we can consider the'Cech cochains determined by this cover with coefficients in the constant presheaf R, a commutative ring, 0* (f R). (For definitions of tech theory and presheaves see [Spanier66].) [Leray50] studied the cochains 0* (f -124; R) within a product structure (of the couverture with the sheaf) that compares with 0* (2.1; 0* ( f (I.; R)), the cochains on X with coefficients in the nonconstant sheaf 0* ( f R). The subsequent filtration induced by weights leads to the spectral sequence. When f is a fibration, f -1 (tIa) F, the fibre, and so we get a system of local coefficients on X. The spectral sequence associated to the filtration has E9 -term identifiable with (X;1:1* (F ; R)), where F is the fibre of the mapping f: E > X. The morass of algebra in [Leray46] was greatly simplified by [Koszu147] who introduced the standard construction of a spectral sequence. The desired applications to the structure of Lie groups and homogeneous spaces soon followed in work by [Leray50], [Koszu150], [Cartan51], and [Bore153]. A careful modern treatment of this construction may be found in [Bott-Tu82]. A discussion of the development of Leray's work is given by [Bore198]. The next development was a spectral sequence of this kind for simplicial singular homology and cohomology, which had been recognized as more versatile tools in homotopy theory. This was accomplished by [S erres 1] in his celebrated thesis. A key technical feature of Serre's work is the determination of the homology spectral sequence. Serre established this result and the multiplicative structure of the cohomology spectral sequence using cubical cochains on a space rather than the classical simplicial cochains. That these multiplicative properties held for simplicial singular theory was proved by [Gugenheim-Moore57] in their thorough study of singular theory applied to fibrations via the method of acyclic models. While we are discussing the various manifestations of classical cohomology, it is natural to ask about analogous results for generalized cohomology theories. Though beyond the intended scope of this book, we mention that the filtration by cellular skeleta of a CW-complex leads to a spectral sequence introduced by [Atiyah-Hirzebruch69] (though known to [Lima591 and [Whitehead, GW62]) for the computation of the generalized cohomology of the complex. Analogous results for a fibration lead to a generalization of this spectral sequence with coefficients in the generalized cohomology of the fibre (see [Dyer69], [Switzer751, or [Prieto79]). For more details we refer the reader to §11.2. Further developments of the structure of fibrations revealed other features of the Leray-Serre spectral sequence. One result in this vein is due to [FadellHurewicz58]. —

223

Other constructions of the spectral sequence

Theorem 6.44. Suppose (F, E,p, B) is a fibration and B is n-connected for some n > 2. Then, in the associated homology spectral sequence, the differentials, di , are trivial for 2 < i < n. Furthermore, for all p> 0, q > 0, dn+1

Hr (B; H,(F; G))

Epn —>E1;±n1 -1,q+n

HP—n-1

(B; Hq+n (F; G))

is given by dn+1 (x) = y n x where the class y is the fundamental class of B in H' (B; and the cap product is defined by a suitable pairing n: 7 +l (B) (F; G) —> H,±„(F; G).

The differential in the theorem is similar in spirit to the transgression—in both cases, a geometric interpretation is found for a particular differential in the spectral sequence. These results were proved independently by [Shih62]. The proof of [Fade1l-Hurewicz58] uses the lifting function for the fibration and the pairing takes the form n: Hri (S213; G) H (F ; G) —> Hq±n (F; G), induced by evaluation at 1 (see §4.3). Another basic viewpoint from which to study the algebraic topology of fibrations is the Eilenberg-Zilber theorem. In particular, does the local product structure of a fibre space mean we can compare C(E) with C(B) and C(F)? In the case of a trivial fibration, we are comparing C. (B x F) with C,,(B) C.(F) and the Eilenberg-Zilber theorem provides a mapping at the chain level that induces an isomorphism 11,,(B x F) II (B) 0 H.(F). For an arbitrary libration, [Hirsch541 generalized the Eilenberg-Zilber theorem by showing, when k is a field, that a differential can be constructed on C"(B; k) 0 H* (F; k) along with a homomorphism C* (B; k) 0 H* (F; k) C* (E; k) inducing an isomorphism, H (C* (B; K) H* (F; k)) H* (E; k). This result provides a smaller cochain complex for H* (E; k). We could also try to provide C.(B) C(F) with a new differential, D, so that H(C.(B) C.(F), D) H(E). Such a generalization was obtained by [Brown, E591. Two pieces of the structure of a fibration come into play: The first is the lifting function for a fibration and the action of QB on the fibre F it provides, p: S2B x F —> F, p,(ct, x) = A(a, x) (1). At the chain level, this induces an action of C(B) on C(B) C(F) C(F). The second piece of structure introduced in G(B) —> C._ 1 (Q13). [Brown, E59 1 is the notion of a twisting enchain, We give the definition in the more general setting of differential homological algebra. Definition 6.45. Let (C, cic) denote a supplemented, differential graded coalgebra and (A, dA) an augmented, differential graded algebra, n: R Co and E : Ao —> R the supplementation and augmentation, respectively. An R-module homomorphisrn C A of degree —1 is a twisting enchain if 3-77 = o = ET and dAr = (-1)"rdc T T where T T is the composite coalgebra

algebra TOT >A. >C0C -- 24.0A

6.

224

The Leray-Serre Spectral Sequence II

The diagonal and Alexander-Whitney maps provide C.(B) with a natural coalgebra structure and the multiplication of loops in QB provides C. (0/3) with an algebra structure. [Brown, E59 1 constructed a twisting cochain in this case using the simplicial structure of singular chains and acyclic models. Suppose (1): (B) —> C._ i (Q.13) is a twisting cochain; this gives a differential on C(B) c(F) as the composite

4:

coalgebra01

> C.(B) 0 C.(B) C.(F) C(B) C C (F) > C(B) 0 C.(S2B) C(F) CC (B) C(F).

10,1,01

The properties of the twisting cochain imply that D = d - 4 is a differential on C. (B) g C(F). The main theorem proved by [Brown, E591 is that the chain complex (C.(B) C(F), d o — 4) is chain equivalent to C(E) and hence has the same homology. There is a filtration of (C. (B) C.(F), D) as a double complex =

q
The associated spectral sequence converges, by Brown's theorem, to H(E). The properties of a twisting cochai ii, along with Theorem 2.15 imply that the the E2 -term is isomorphic to H.(B; (F; R)). Brown carefully compared this spectral sequence with Serre's to prove that they are isomorphic. Furthermore, Theorem 6.44 is a corollary of this comparison of spectral sequences. The notion of a twisting cochain was introduced by [Cartan541 as the algebraic analogue of the simplicial twisting functions introduced by [Moore561. The interested reader can consult the book of [May67] or the paper of [Husemoller-MooreStasheff741 for more details. An example of an explicit spectral sequence calculation using this level of control of the first differential can be found in [McCleary90]. An algebraic development founded on these ideas is known as homological perturbation theory [Gugenheim-Lambe-Stasheff91]. The main idea is to generalize the relation between de and de — 4 on C(F) C(B). Given a tensor product of differential graded modules, (M, dm) 0 (N, d 0 ). there is the tensor differential on M 0 N given by de = dm 0 1 + (±)1 0 dN. A perturbation of d with respect to a decreasing filtration F. of M 0 N is a sum D= de + d2 + d3 + • • • ,

with ds (Fk) C Fk_ 8 . The condition DoD = 0 places a strong restriction on the homomorphisms di. When a perturbation exists there is a spectral sequence with E2 -term given by H(M ®N, de ) and converging to H (M N, D). Furthermore, the differentials in the spectral sequence are closely related to the di in the sum. [Gugenheim60] gave this formulation of the results of [Brown,

Other constructions of the spectral sequence

225

E591 and [Hirsch541. Since then a very elegant and satisfactory theory has been

developed. See the papers of [Lambe92] and [Huebschmann-Kadeishvili91] for recent reviews of the subject, as well as the work of [Chen771 and [Gugenheim821. Bisimplicial sets and Dress' construction 0

The idea of a locally trivial product with a global twisting is realized algebraically in the twisted tensor product (the tensor product with the new differential). We next describe a more geometric construction that leads to the Leray-Serre spectral sequence by exploiting the local product structure. The construction is due to [Dress671. The double complex involved was introduced in [Fadell-Hurewicz58] without the explicit simplicial structure. There is also a nice discussion of this work to be found in [Liulevicius67]. Our point of departure is an object that carries the local product data of a fibration in an fashion analogous to the way that the simplicial structure of a space carries its homology. Definition 6.46. A bisimplicial set, X., is a bigraded collection of sets, X p ,,, for p,q > 0, together with vertical and horizontal face and degeneracy X p , q _ i , and maps, Xp.q Xp-1,q, e5 1 1 . Xp,q Xp+1,g, a: si' : X,,, X,,, +1, indexed over the usual sets, commuting with each other and satisfting the standard simplicial identities in each direction. A bisimplicial set may also be defined as a simplicial object over the category of simplicial sets (a simplicial simplicial set). The fundamental example is the bisimplicial set A.. with Apq = A P X Aq and 0, 8/1, 61"t,', sT, the usual face and degeneracy maps. To an epimorphism of spaces, f: E B, we associate a bisimplicial set E, v: AP -> K..(f) as follows: Let Kpq (f) = {(u,v) u: AP x Aq B continuous with f ou= y o py1 }; here pri is the first projection mapping. Thus (u, y) is in Kpq(f) if the following diagram commutes A P x Aq

)E

13 r

AP

)

B.

Let (u, v) be in Kpgr(f). The face maps : Km(f) —> Kp _i,Jf)

and d;', : Km (f) —> Kr,q_i(f)

6.

226

The Leray-Serre Spectral Sequence II

(u, y) and a(u, I));

are described by the diagrams for (U,

V)

:

A P-1 x

1

Pr

A P-1 19/V,

) A P x Aq

Ei

Aq -1 1 XEk

(U.

i

pr

Pri

)

AP

AP X

B

V

Aq

u

E

If

Pri

AP

If

B.

AP

Here ei denotes the inclusion of the jth-face of the image simplex. The degeneracies are defined similarly. Apply the free abelian group fiinctor TA to K.. (f) to get a bisimplicial group that we denote by Spq (f) = ,FA(Kpq (f)). This bisimplicial group determines a double complex by taking d' : SIN( f) Sp_ i ,q (f) and d" : S pq(f) Sp, g_1(f) to be d' =

7=0

(-1) 3 ,FA(0 / )

and d" =

k=0 (-1)q±k'TA(an.

The signs and the simplicial identities conspire to give d' o d' = d" o d" = d' o d" + d" o d' = O. When we apply the analogue of Theorem 2.15 for such double complexes, we obtain two spectral sequences. Theorem 6.47. For the double complex {S.,.(f),d',d"}, the following hold;

_ d"), d' )

{H q (E),

if p =

If f is a fibration with fibre F, then

Hp (Hq (S.,.(f), di ), d") Hp (B: li g (F)) where 7-14 (F) is the local coefficient system given by the homology of the fibre over each point in B.

Thus the double complex (S.,. ( f), d', d") gives rise to the Leray-Serre spectral sequence. This approach provides considerable simplicity of the constniction of the spectral sequence as well as a framework for studying the spectral sequence for various chain and cochain functors that support classical homology and cohomology, for example, de Rham cohomology (see [Grive1791

Other constructions of the spectral sequence

227

and [McCleary821). The argument given here can also be generalized to obtain the spectral sequence for fibrations of pairs aLiulevicius671). PROOF: We display the double complex S* , * (f) as follows

820

(f)

810

(f)

<

821

(f)

811

(f)

S01(f)

822(f) "<-

823(f)

<

S12(f)

<

813(f)

<

S02(f) <

S03(f)

d'

Soo (f)

First consider H* (Si* (f d"). This is clearly the homology of the set Ki* (f). Furthermore, H* (So* f), d") = 11* (E) follows from the definition of Ko q (f) and that of the singular complex for E. To establish the first part of Theorem 6.47, we examine the complex of graded abelian groups: (90. (91.+92. H* (S i„(f), d")< TI*(S2*(f), d') <— —

I-1* (So* (f), d")<

Lemma 6.48.

For all j > 0, K0(f) K3*(f).

PROOF: In fact, we prove a bit more by showing that there arc mappings : Ko, * (f) K. (f) that are homotopy equivalences and that the following diagram commutes up to homotopy Ko,* (f) Ji

Kii,*(f)

—

Ko,* (f ) Ji_1

-K —1,*( f).

The mappings, Ji , are induced by taking the collapse map Ai —> which is a homotopy equivalence with homotopy inverse A° —> Ai defined by taking A° to the barycenter of &. The maps induced by the collapse and its inverse give a homotopy equivalence. Furthermore, this homotopy equivalence is chosen —> Ai+ 1 that induce the simplicial to be compatible with the face maps r: structure on K..(f). It follows that the homotopy equivalence induces a mapping of simplicial spaces:

6.

228

The Leray-Serre Spectral Sequence II id

id

Ko *

id

J2

id

KO* =1,10

J1

t

00

id

Ko *

0 01

K2 *

K.

00

K0*(f).

The mappings Ji induce the identity on 11„ (E) for all i and so mappings induced by a, may be taken to be the identity for all i. Applying the result of the lemma, the graded complex that computes II* (S** (f), d") can be identified as the complex: 0

id

H(E) H(E)

0

id

H(E) H(E) <— • • •

and so part (i) of the theorem is established. We sketch a proof of the second half of Theorem 6.47. Examine a (p, q)simplex, (u, y).

This identifies a p-simplex of B and a family of q-simplices of f'(v(x 0 )), parametrized by points xo in A P Since AP is contractible to its leading vertex, if f is a fibration, then f'(v(x0)) is homotopy-equivalent to f -1 (v(eo)) = Fv ( eo ), the fibre over y(e0), the image of the leading vertex of AP. Fix V: AP B and consider the set {chains Ei ai (ui , y) E Spq (f), for any ql. Each , y) identifies a mapping, Aq —> f -1 (y(e0 c AP)), and so the homology, with respect to d', of this set over y is H* (f'(y(e0 E AP ) Each y and each (ui , y) contribute an generating chain to cp(B ; 7-I. (F)) (see §5.3 for the relevant definitions) and so we have established that H (S* , * (f), d') .

C* (B;71 * (F)).

To complete the proof, it suffices to show how d" induces the differential for chains on B with local coefficients. To do this, one must keep track of the leading vertex of an elementary chain on B through the identifications and boundary maps al'. When the leading vertex does not map to a leading vertex, however, the identification of the coefficients with H* (F(,)) is carried across the simplex along the path given in the definition of homology with local coefficients (see §5.3) and so d" determines the correct differential. This establishes the second half of the theorem. Hiress671 identified the notion of systems of local coefficients with certain functors on the category of simplicial objects over a fixed space. The details of his proof provide a combinatorial description of local coefficients.

Exercises

229

We have discussed several constructions of the Leray-Serre spectral sequence. We mention one other method of construction of the Leray-Serre spectral sequence: [Eilenberg50] and [Damy96] gave constructions based on the use of a Cartan-Eilenberg system. The fact that these different constructions give rise to the same spectral sequence reflects the restrictions imposed by the structure of a fibration. These restrictions can be made very precise. [Barnes85] introduced axioms that a functor F, from the category of fibrations with all spaces locally finite CW-complexes to the category of filtered chain complexes, must satisfy in order to give a spectral sequence like the LeraySerre spectral sequence. Such a functor is called a fibration spectral sequence constructor. He shows that the constructions we have discussed fit into this framework. Furthermore, there is a comparison theorem for natural transformations between constructors t: F —> F' that implies when two such functors give rise to the same spectral sequence. There are notions of a free and cofree functor, for which there are always natural transformations and the construction of [Dress67] is a free functor. [Barnes85] proved the following uniqueness result via an acyclic models argument. Theorem 6.49. Let F and F' be two fibration spectral sequence constructors. Then there is a canonical isomorphism t: —> for all r > 2, of the associated spectral sequences as functors on the category of CW-fibrations.

The Leray-Serre spectral sequence takes information about the fibre and base of a fibration and proceeds to information about the total space. As we have already seen in Theorems 6.19 and 6.38, we may want to go from data about the fibre and total space to the base (for example, the cohomology of classifying spaces) or from data about the base and total space to the fibre (for example, the cohomology of OX). In the most successful examples (Eilenberg-Mac Lane spaces and classifying spaces) the answer depends heavily on the manageable form of the input—for example, when there is a p-simple system of generators. Then the output takes the form of a homological functor on the input. How systematic is this algebraic phenomenon? Can it be generalized? There is a unified approach to these computations, and once again, it comes in the form of a spectral sequence. The construction, due to [Eilenberg-Moore66], and its applications are the subject of the next two chapters.

Exercises 6.1. We used certain facts about Ext in the computation of 74 (S2 ). Prove that Ext (7, Z)

7

when 7 is a finite group.

6.2. Show that the transgression (Definition 6.3) is well-defined.

6.

230

The Leray-Serre Spectral Sequence II

E -is a libration and fi* (E; R) {0}. Show that Hr (F; R) = {0} for 0 < j < q if and only if Hj (B; R) = {0} for 0 < j < q ± 1 and furthermore, the transgression, T: Ht (F; R) —> Hr*" (B; R) is an isomorphism for t < 2q. 6.3. Suppose F

6.4. Show that the transgression annihilates products. 6.5. Prove the following theorem of [Bore1501: Suppose k is a field and F E —> B is a libration with H* (E; k) H* (Sri: k). Suppose further that H* (F; k) H' (S2m 4+ 1 X 52m2 + 1 y... X S2 ''+ 1 ; k), the cohomology of a product of odd spheres and B a finite complex. Then s = 1. 6.6. Prove Theorem 6.8. 6.7. Compute 75 (S2 ) by the method of killing homotopy groups, that is, using the associated upside-down Postnikov system. 6.8. Prove Lemma 6.18. 6.9. Complete the proof of Theorem 6.19 (theorems of Cartan and Serre) by computing H* (K (II, n); Fp ) when H = E/pk 2. Also prove the mod 2 version of Theorem 6.19. 6.10. Use Theorem 4.43 and the results of Cartan and Serre in §6.2 to show that the Steenrod algebra, as described in Chapter 4, contains all stable cohomology operations over 7-1 Ip7Z 6.11. Show that the transgression, r: Hn-1 (Sn-1 ) —> Hri (Sri), for the fibration, Sn -1 To Sn —> Sn, the sphere bundle associated to the tangent bundle, is given by multiplication by the Euler number of Sn. 6.12. Generalize the Hopf algebra argument of the proof of the Hopf-Borel theorem: Suppose A C H is a sub-Hopf algebra of a given commutative Hopf algebra H. Let M HIIA be a sub-vector space of M such that q : M —> H//A, the restriction of the canonical quotient map, is an isomorphism. Show that A M is isomorphic to H as left A-modules. (Hint: Consider the composite mapping

A

M

inc O (qI )

HOH ±>H—> HOH /

leg >H0(HIIA).

Show that the composite is a monomorphism. Show also that A together with M generate H as an algebra, so the composite of the first two maps in the display is onto.) 6.13. Prove that the Whitney sum formula follows from Lemma 6.41. 6.14. The axioms for Chem classes are given for e = (C n , E,p,M), a complex vector bundle over the base space M, as follows: I. There are classes ci() G H2' (M) and co (e) = 1, c2, (e) = 0 if j > n.

Exercises

231

> M2 is a continuous mapping that is covered by a bundle map, that is a linear isomorphism on th e fibres, then c2( l) (f f): j —>

II. If f: M1

—

,

III.

f * (c 2i (6)) • For two vector bundles,

and n , the Whitney sum formula holds:

c2k(e ED 77) ==

c2i(e)

i±j=k

2j(7)).

IV. Let = ( cl, E,p,CP(1)) be the canonical complex line bundle over CP(1). Then c2(74) O.

Prove the analogue of Theorem 6.22 for Chem classes and H* (BU(n)). You can assume that the analogue of Lemma 6.23 holds for the mapping BUN x BU(m)

—

> BU(n m).

6.15. To prove the analogue of Lemma 6.41 for BU(n), one uses the maximal torus in U(n), Tn S 1 x • • • x S l (n times). The inclusion of Tri in U(n) induces an isomorphism,

H* (BU(n)) = { Symmetric polynomials in

, y, deg y • = 2 }.

The result still holds mod p and H* (BTn; Fi) ) has a particularly simple Ar -structure. From this, deduce the Ar-structure on H* (BU(n); Fr). Do the analogous computation for H* (BSO(n); 72).

6.16. Compute H* (C) ( 2m) I u (m );7 2) and H* (U(n) I Tn).

7 The Eilenberg-Moore Spectral Sequence I the application to topology of homological algebra leads to somewhat different developments which may be included under the heading of differential homological algebra."

.. . ,

J. C. Moore

The Leray-Serre spectral sequence provides a method for computing the cohomology of the total space of a fibration from knowledge of the cohomology of the base space and the fibre. By arguing backward through the spectral sequence, the inverse problems of computing the cohomology of the fibre (as for the path-loop fibration) or the cohomology of the base space (as in the case of classifying spaces or Eilenberg-Mac Lane spaces) from the cohomology of the other two spaces in the fibration can sometimes be solved (Theorems 5.16, 6.20, and 6.39). In the particular case of the computation of H* (BG; k) from H*(G; k) when G is a compact Lie group, the algebraic relation between H* (G; k) and H* (BC; k) is often expressible in the language of homological algebra and derived functors. In pioneering work, 1Cartan541, [Moore59], and [Eilenberg -Moore661 developed the correct algebraic setting to explain this relation. We present in this chapter the homological framework that leads to a general method of computation. To begin, we extend the problem of computing the cohomology of the fibre from the cohomology of the base and total space to a more general question. Suppose 7r : E .13 is a fibration with fibre F and f : X —> .13 is a continuous function. A new fibration, 7r' : Ef > X, can be constructed from these data, namely the pullback (§4.3) off:X—>B<—E:7, as in the diagram —

233

7.1. Differential homological algebra

Problem: Compute H* (Ef; R) from knowledge of H* (E; R), H* (B; R), H* (X; R), f* and 71 *, where R is a commutative ring with unit.

This problem includes the problem of computing H' (F; R) from H' (B; R) and H* (E; R): Let 1)0 be a basepoint for B and b0: * —> B, the unique map in Top that selects the basepoint. If we take f = bo in the pullback diagram, then Ef = F and the initial data are H* (E; R), H* (B; R) and 71 *. The solution to the problem comes in the form of a spectral sequence, introduced by [Moore591 and developed by Eilenberg -Moore661. Theorem 7.1. Let 7F : E B be a fibration with connected fibre for which the system of local coefficients on B determined by 71 is simple. Let f: X —> B be a continuous function and E f be the total space of the pullback fibration of 71 over f. Then there is a second quadrant spectral sequence, {E'* ,d,}, converging to H- (E f; R), with E2"

Tor*'* 11. (B;R) (H* (X ; R), H* (E; R)).

When B is simply-connected, the spectral sequence converges strongly.

New algebraic func tors, Tor*Ri: (B , R) (— , —) and Tor*(g* (B ;R) were introduced by [Moore591 that generalize the derived functors Tor R (—, —) of the tensor product to the category of differential graded modules over a differential graded algebra. The cup-product structures on H* (X; R) and H* (E; R) together with f* and 7*, respectively, provide each with an H* (B; R)-module structure. The bulk of this chapter (§7.1) treats the algebraic foundations that support the construction of the spectral sequence of Theorem 7.1 and make it calculable. In §7.2 the algebra is translated into topology. In §7.3 we further streamline the homological computations in special cases using the Koszul resolution. In this context we compute again the cohomology of the complex Stiefel manifolds (compare Example 5.G). The Stiefel manifolds are a special case of the computation of the cohomology of a homogeneous space: If H is a closed connected subgroup of the Lie group G, then GIH may be taken to be the fibre of the mapping B(inc): BH —> BC induced by the inclusion. Theorem 6.38 provides H* (BG;k) in many cases and so Theorem 7.1 may be applied to compute H* (GI H;k) (see §8.1). We also consider the computation of the cohomology of the based loop space of an H-space. In §7.4 we discuss the dual case of a pu shout diagram and the EilenbergMoore spectral sequence for homology. This is especially useful in computing H* (BG;k). We postpone a complete discussion of the problem of strong convergence of the Eilenberg-Moore spectral sequence to Chapter 8bis . More sophisticated applications and the development of the deeper structure of the spectral sequence are taken up in Chapter 8. The reader with little exposure to homological algebra can consult §2.4 for a primer on the homological algebra that is generalized in this chapter. (— —)

7. The Eilenberg-Moore Spectral Sequence I

234

7.1 Differential homological algebra

Let R denote a commutative ring with unit and let C* (Y) denote C* (Y; R), the singular cochains on Y with coefficients in R. We begin our discussion of homological algebra with the motivating topological example. The Alexander-Whitney map provides C* (Y) with a multiplication, C(Y) oR C(Y)

for all p, q. The differential, 6: CP (17 )

C(Y) CP+1 (Y),

satisfies the Leibniz rule,

6(a ---- h) = 6(a) ---- b+ (-1)deg a a ---- 6(b), and so (C' (Y), 6) is a differential graded algebra over R. We can apply the cochain functor to a pullback diagram

to get the commutative square: Ef

ir

1 X

C* (E f)

,E

f

C*( ;R)

1

C* (X) ,

> B

C* (E)

f*

I '7

C* (B).

The morphisms of differential graded algebras, f" and Tr * , endow C* (E) and C* (X) with C* (B)-module structures: The (right) module structure map for C* (X) is given by cp : C* (X) OR C* (B) C* (X), which denotes the composite C*

(X) OR C* (B)

10f*

C* (X) OR C * (X)

C* (X)

C* (E) denotes the and similarly, =— o(71- * g 1): C*(B) (DR C* (E) (left) module structure map for C* (E). We take these module structures as the basic data. The space Ef associated tof:X—>B<—E: 7 enjoys the universal properties of pullbacks. It is natural to ask whether or not C* (Ef) shares any dual universal properties in the category of C (B)-modC* (X) 0c * (13) C* (E) ules. A first guess for C* (Ef) might be C* (Ef ) (with the right hand side properly defined to generalize the case of the tensor product of two R-modules). We will see later (Lemma 7.3) that this implies H* (E f; R) H* (X; R) H * ( B;R ) H*(E; R). If we apply this guess to the path-loop fibration, it fails to compute the cohomology of the based loop space. Notice, however, that when B = *, the guess above is correct. A more sophisticated second guess can be made by observing that the first guess works for B = * because C* (X) and C* (E) are free C* (*)-modules. (This is the Kiinneth theorem.) Furthermore, if 7 and f are projections, the same argument works. The deviation from "freeness" of C* (X) and C* (E) as C- (B)-modules reflects the nontrivial twisting of the base and fibre as a product in the fibration, E B, and further as pulled back over the mapping f: X —> B. In the case of modules over R and differential graded modules over R (see

7.1. Differential homological algebra

235

§2.4), this information is expressed through the derived functors TorR(—, —) of the tensor product. The correct expression for H*(E f ; R) in terms of the E: 7F is based on a generalization of the data provided by f : X —> B theory of differential graded modules over R to a theory of differential graded modules over a differential graded algebra over R. In our exposition, we follow [Moore59] and [Smith, L67 1 .

Differential graded modules over a differential graded algebra

Let (r, d) denote a differential graded algebra over R. For completeness, we recall Definition 1.4:

(1) r = e n>0 r- is a differential graded module over R. We take d, the differential, to have degree +1. (2) There is an R-module homomorphism, m: r OR r —>r, the multiplication, which is associative, that is, the following diagram commutes: F F

r

mel

F OR F

10m

(3) d is a derivation with respect to m, that is, if we write m(a then d(a • h) = d(a) • b (_ i )cleg a a • d(b) .

(4) r has a unit, that is, there is an injection 77 :

R

b) =

a • b,

r of rings, so that the

following diagram commutes

r oR R

10 97

r eRr< n ®' Ro R r

The ring R may be taken to be the differential graded algebra, (R, d = 0), with R in degree zero, {0} in higher degrees, and the zero differential. Those differential graded module morphisms, : (F, d) —> d'), of degree zero that commute with the multiplications and the units comprise the morphisms of differential graded R-algebras and so determine a category, denoted DGAlgR . Singular cochains on a space with coefficients in R and the cohomology of a space (with zero differential), both with the cup product, are functors Top —> DGAlgR from spaces to differential graded algebras. A left module, N, over r is a differential graded module over R, together N, which is a morphism of differential with an action of F, F OR N

7. The Eilenberg-Moore Spectral Sequence I

236

graded R-modules such that the following diagrams commute:

r el? r eR N

r eR N

10n1

r eR N

r oR

N.

A morphism of left r-modules, p: N —> , is a homomorphism of differential graded R-modules that is compatible with the F-actions. The left (differential graded) F-modules constitute a category that we denote by DGrMod. The category of right (differential graded) r-modules, DGModr, is defined

similarly. Suppose M is a right F-module and N a left F-module with structure maps and 0, respectively. We define M or N, the tensor product of M and N over r, via the short exact sequence in DGModR

eR r oR N

so01-1evi

>M OR N M

or N

O.

Thus M or N can be described as M (SRN modulo the relations (m • y) on = m (-y • n), for all -y E r, M E Al, and n E N. For N in DGrMod, the unit in r may be used to construct an isomorphism between N and r Or N Or N as left F-modules: Consider the R-module homomorphisms f : N given by f (n) = 1 or n, and g: r ®j N —> N given by g(-y or n) = -y • n. By the definition of or, f and g are inverses. We consider F Or N as a left F-module with the action -y (y' or n) (y7') or n. The identification (-y-y') • n = y • (y' • n) is equivalent to the fact that f and g are F-module homomorphisms. Thus N r or N as left F-modules. By the symmetric argument for a right F-module M, M r M as right F-modules. We next define the derived functors of — Or —, for which we need a notion of exact sequence. Let •••

(Qn ,dn )

(Q n+1 c1,2+1) —> (Qn+2 cln +2) —> • • •

be a sequence of morphisms of left F-modules. Recall from the case of differential graded modules over R (§2.4) that the sequence is proper exact if it satisfies (Qn ) 4 (Qn+1) # (Q n+2) 4 • • • is exact in the category of graded F 4 -modules, where ( : DGrMod —> GrMod is the forgetful functor that ignores differentials. 1) (2) Z(Q) Z(Q Z(Q 2) is exact in the category of graded R-modules, where Z(Q) = Z (Q n , cl„) = ker dn • —> H(Qv+ 1 ,c1 ±1) H(Qn+2 ,c17,4 2) —> • • • is (3) • • • —> H(Qn , exact in the category of graded R-modules. (1) • • •

237

7.1. Differential homological algebra

If N is a left F-module, we define a proper resolution of N to be a proper exact sequence in DGr Mod: 6

• • • -> Q -n

6

Q -n+1

6 6 -> • • • ->

Q -1

6 ->

Q0

N

Having defined the notion of exactness, there is a corresponding notion of projective. We say that a F-module, P, is (proper) projective if, for any morphism, o- : P N and proper epimorphism M N in DGrMod, there is an extension, 6- : P M in DGrMod so that o 6- = o- :

N

) O.

We observe that the category DGrMod has enough projective modules. Proposition 7.2. Suppose r and H(r) are flat modules over R and N is a left r-module. Then there is a projective r-module, P, and a proper epirnorphisrn, 7r: P —) N. That is, every left r-module is the image of a proper projective r-module.

PROOF: We first introduce the notion of an extended module. Suppose V is a proper projective differential graded R-module; then V, Z(V) and H (V) are all projective modules over R (see §2.4). Consider V = r OR V and give V the left F-module structure, cv : r OR I-7 - —> 1-'7 as in the diagram av

F

r OR OR

V

r OR V.

V is a projective F-module. N is a proper epimorphism and a: V Suppose (: M N is a (left) F-module homomorphism. Observe that V sits in V as 1 R)R V since F has a unit. We claim that

238

7. The Eilenberg-Moore Spectral Sequence I S

Because is a proper epimorphism, M —> N —> 0 is exact in DGModR and so we can extend al v : V —> N to an R-module homomorphism v : V —> M. Extend I v to all of V by 6- (-y ® = 7 • a-l v (0. Since a is a r-module homomorphism, 6- is our desired extension. To complete the proof of the proposition, we use the fact that there are enough projective differential graded R-modules and take 7r0: P° —> N, any proper epimorphism in DGModR . Extend 7ro to the r-module homomorphism 7T : r ® Po N, which is the composite 107ro :

N

R

It remains to show that 7T : r

(P

N.

N is proper epic.

Po

Lemma 7.3. Suppose that r and H(r) are flat modules over R, P an extended (hence projective) left F-module and B a right F-module. Then H(B or P) H(B)O H (r) H(P).

PROOF: With the assumption of flatness, the Kiinneth theorem allows us to give H(B) an H(11-module structure by H(B)0 R

H(r)

H(Bo R

r)

H(PB)

> H(B).

Since P is an extended module over r, we can write P = r OR V and compute, applying the Ktinneth theorem at the appropriate points, H(B or = H(B or

r ®R v)

H(B) oR HO/ H(B)

H(B ®R V) H(B) H (F)

0„ (,) H(r ØR v)

R H(V)

H(B) Ow) H(P).

We define a proper projective resolution of a left r-module, N to be a proper exact sequence • • • —>

•••

p-1

p

0

6

N

0

for which each P–i is a proper projective l'-module for i > O. Lemma 2.19 guarantees that proper projective resolutions can always be constructed.

Corollary 7.4. Suppose

r and H(r)

are flat modules over R and

p—i+1

P-1

PO N

is a proper projective resolution of N, then H(S) —> (P

H(s)

H(6) —> (P

H(6)

H(6) (P1

H (N)

7.1. Differential homological algebra

239

is a projective resolution of H(N) as an H(r)-module.

PROOF: By Lemma 7.3 we may take all of the P—i to be extended modules over F. Since r is flat, H(P) H(r) H (V —i ) by the Kiinneth theorem. We take V —i to be proper projective, so H(V) is projective over R. Thus H (P—i ) is an extended module over H (1). since a proper projective resolution is proper exact, and H is flat, the sequence on homology is long exact. Hence we have an H (F) -projective resolution of H(N). As in Chapter 2, we have chosen the homological grading for the resolutions in such a way that, with the internal degree, we have a double complex:

>

(T)-1 2

p0)2

> N2

(T)-1)1

( .p0)1

N1

jc1_,

jcio

(T)-1)0

( .p0)0

> (

> 13

I dN NO

O.

Suppose (P• , 6) is the complex p-2+1

p-1 p0

This complex has homology H (P• , 6) = (N, dN ). Form the Z-graded differential module, (total(P*), D) by letting

=EE m+n-j

[total(P')]

D

=

(6+ (-1)" dm ).

(We write the direct sum with the word of warning that, in the general case, this sum may be infinite and so the weak direct sum is intended. In certain situations of topological interest, such large sums do not occur.) There is a r-module structure on (total(P•), D) given by restricting the F -action, r OR to (P)m+. Since the action determines a mapping rk OR (P — i)+i (P — i )m +i+k, we get rk [total(P•)]m [total(P')]m+k . If M is a right r-module, we form the E -graded differential R-module (M

or total(P.), dm 1 + (

1)deg D)

where the sign is given by dm 0 1 + (-1)(1 1 (s D on elements represented in Mg

total(P').

7. The Eilenberg-Moore Spectral Sequence I

Definition 7.5. Tarr (M N) = H(ill

total(P"), dm 1 ± 1

D).

We leave it as an exercise (compare Proposition 2.17) to verify that the definition is independent of the choice of proper projective resolution as well as independent of the choice to resolve N in DGr Mod and tensor on the left with M or to resolve M in DGMod r and tensor on the right with N or resolve both and tensor the resolutions together. (In the topological applications to come, we will resolve right modules.) Furthermore, the reader should verify that Torr is the correct generalization of TorR, in the sense that Torr (M, -) behaves as the left derived functors of M We make the following useful observations: (1) Torr (M, N) is bigraded.

This follows by writing Torr-i'* (M, N) = =

or total (P*), e 1 + ( - 1)deg D) a subquotient of M er P.

Those elements coming from M'n or (P- ) n have bidegree (-i, rn + n). The first degree is the homological degree and the second is the internal degree.

(2)ED Tor°4 (1t1 N) = M or N. This follows from the fact that If(1", 5) = (N,dN ). Thus Torr (M N) contains not only M or N but also the deviation from exactness of the fiinctor M Or - and the deviation from being projective of N. In the topological interpretation, Torc.(B)(C'(X),C* (E)) measures the twisting effect of f* as the deviation from exactness of C(X) 00.(B) — and the nontriviality of as the deviation from freeness of C* (E) as a C* (B)-module, (3) total(P) is filtered.

An increasing filtration of total(P) is a consequence of having a double complex and so we get the filtration: {0} = F1 c F° C F -1 C • • C F C • • c total(P)

EE

j> -n. defined by (F-n)r = up where the sum is over i + j = r and We can picture the submodule F- n as in the diagram:

7.1. Differential homological algebra

241

It is immediate that D: F —> F– n and each F' is a E-graded r-modF–n . By Theorem 2.6 this ule. We filter M or total(P) by F– n = M filtration leads to a spectral sequence. Theorem 7.6 (the first Eilenberg-Moore theorem). Suppose r and H(r) are flat modules over R. Then there is a second quadrant spectral sequence with E;'* rforW(r) (H(M), H(N))

and converging to Torr*'* (M, N).

PROOF: Having set up the algebra carefully, we use Theorem 2.6 to get most of the theorem. It suffices to compute the E2 -term. The initial term of the spectral sequence is Er ,* n .F– n+ 1 ) and, as in the proof of Theorem 2.20 (the Kiinneth spectral sequence), we have E* H* (,F– n / T –n+1 )

H*(M

By Lemma 7.3 we obtain E,'" H(M) ® H (r) H(P) and di = 1 ®H(). Corollary 7.4 and the definition of Tor give us the desired E2-term. As in the discussion of Theorem 2.20, the strong convergence of this spectral sequence may be delicate—the stages in the filtration are Z-graded. A judicious choice of resolution can sometimes be made that insures strong convergence (for example, in the discussion to follow on the bar construction). General convergence results are discussed in Chapter 8b1s . An important corollary of the theorem is due to [Moore591 that motivates a number of results in Chapter 8. Corollary 7.7. Suppose r and A are differential graded algebras over R, M is a right F-module, N is a left F-module, M' is a right A-module, and N' is a left A-module. Suppose we have homomorphisms, f: r A, g: M —> M' and h: N —> N' satisfying (1) f: r -> A is a homomorphism of differential graded R-algebras. (2) The following diagrams commute

moRr 9®

,PM

is

fl,

M' g R

M

A

ww ,

M'

r 0,, N

f0

it

A o R N'

(3) H(f), H(g) and H(h) are all isomorphisms.

,»N h

N'.

242

7. The Eilenberg-Moore Spectral Sequence I

Then, when the associated spectral sequences converge strongly,

Tor f(g,h) : Torr (M, N)

TorA(Mi , N')

is an isomorphism.

PROOF: The corollary follows directly from Theorem 3.9 and the fact that TorH(f )(H (g), H (h)) is an isomorphism of the E2-terms of the associated spectral sequences. This corollary indicates how Tor, under suitable conditions, "sees" only the homology of the objects involved. The subtleties of what is lost in this procedure figure prominently in the applications of this spectral sequence to the computation of the cohomology of homogeneous spaces. At this point we should introduce the topology and interpret this algebraic spectral sequence for topologists. The impatient reader can look ahead to §7.2. However, before we leave the algebra, it is useful to recast our results in a (theoretically) computable rather than categorical context. The bar construction

In order to compute Torr (M, N), we need a proper projective resolution of the left f -module N. Since any proper projective resolution will do, a constructive, functorial complex is desirable that perhaps carries some extra structure that can be exploited in calculation. One of the most useful explicit constructions in homological algebra is the bar construction, which was introduced by [Eilenberg-Mac Lane531 in their study of the cohomology of abelian groups. For the sake of simplicity, we assume that our commutative ring R is a field k. The interested reader can extend the construction (led by the Kiinneth theorem) to arbitrary rings (see the exercises). Suppose that f is connected, that is, the unit, : k —> f is an isomorphism in degree zero. Denote the cokernel of n by F. Since r is connected, we have that f = r+ = {-y E F deg -y > 0 } . Define n times 13 —n

(r,N) = r (x) k r o k • • • g k r

Notice that 13 – "(r. N) is a left f -module with the extended module action. It is customary to write an element of B –n(r,N) as -y[-yi 1/2 " -Yn]a and to write -y [ ]a for elements of B ° (r, N). To simplify signs in many formulas, we adopt the following Convention: If a is an element of a differential graded module, then (

1)1±cleg act.

243

7.1. Differential homological algebra

The internal differential d' defined on 13 -n (r, N) takes the form cr n (- [ 1 • • • I -rn] a) = (dr -y)[ -yi

I

I -yr ] a, +

•••

i=1

I

•701

"Y

•••

I "Yi-1 dr7i

I

-yi+i

I -rnia

•••

n](dNa).

•••

We leave it to the reader to check that d- n is F-linear. We assemble the differential graded F-modules (13 -n (r, N), d) into a resolution by introducing the homological differential 6: B --(r,N) 13 -n±l (r,N). ( -1 ) deg7 6 (7[711 • 7n]a) = 70172

I

•••

n-1

I -ynla +

-yrryi +-yryi

I

•••

I • • • I "Yi-i I "Yn-ii(eYna).

• 7i+i • • • 7nict

(Our sign convention follows [May67] and agrees with [Smith, L67 1 . Though more complicated than Mac Lane70], it simplifies some notation used in §8.2.) The reader can verify easily that 66 = 0, 6 is F-linear and d'6 + 6Crn = O. If we let r = N : B° (r N) = r g k N N, then we have the complex of F-modules • • • ->

(r is

(5

B -n(r,

(5

—> • • • —> B -1 (F,N) —>

B°(r,

—> N —> 0

onto, because r has a unit).

Proposition 7.8. (13 * (F, N), 6, e, r) , as a complex of differential graded rmodules, satisfies

(1) H(13 . (r,N),6,e) = {0}, that is, the sequence is exact. (2) H(B - n(F, N), d-n ) = B - n(H(F), H(N)). (3) (13 . (r, N), (5, r) is a proper projective resolution of N over F.

PROOF: Consider the contracting homotopy: '••

1 [7 71 ' • • 7nia, 7n]a) = {

0,

if deg -y > 0, if deg -y = 0,

and s(a) =1[ ]a. We verify Ss + s6 = id: 6s(7[7i • • • 7n]a) = 6 ( 1 [7

I

I

•••

I 7rda)

-1 = 7[71 • • • 7n]a +1rry

I

I

•••

7,+1

1 ["Y "Yi • • •

I 'Yn-11( -rn0) +

I

••

I

I

•••

-rn]a.

I

N]a,

7. The Eilenberg-Moore Spectral Sequence I

244

Next we compute sa: 6 7 [-y,

8 (

• • • Nict) =

s

( - 1-n[72 • • •

I

7n1a) —1/ryi

n-1

—s(

), [)/1

2=1

= —1 [111

I 72 I 1['ry

I

),]_

• • •

• • •

I 'Yn—i](eYna))

I 'YieYi+i

•••

7n]a

7n]ct — 1 [1" 1'1 • • •

• • •

—1 i=1

I

I

I

• •

I 'rYieYi+i

•••

-7,1 _1]( -yn a)

eYnict.

Combining the two displays we find that Ss + s6 — id when n > 1. For the bottom of the resolution, observe that s(a) = 6 (1[ ]a) =a and sa(a) = s(0) = O. It follows immediately from Ss + sa = id that the complex is exact. Observe from 6d —n d—n+1 6 = 0 that 6(Z(B'(F, N), d)) c Z(B" ±i (r, N), d

e ').

The reader can verify that sd— n + Cr n— S = 0 with the consequence that s carries ker d into ker . If we restrict s to Z(B . (r, N), d), it remains a contracting homotopy and so the sequence of graded R-modules

z(B --(r, N), d') L Z(B —T1+1 (r,N), d-n+i) L • • • • • • —>

Z(B -1 (r,N),d -1 ) —> Z (B ° (1 , N), do)

Z(N,dN)

0

is exact. Furthermore, when we ignore the internal differentials, the complex remains exact because 6s + sa = id does not involve the internal differential. This proves (1). To complete the proof of (3), we establish (2). From the Kfinneth theorem and the fact that k is a field, (2) follows from n

times

H(13 -71 (F, N), d—n ) = H(F) o k H(F) ok • • • o k H(F) o k H(N) = B'(H(F),H(N)).

Passing to homology, Ss + sa becomes H (6)H (s) + H (s)H (6) = id and the sequence of homologies of the B' (r, N) is exact. The proper projective resolution of N over r, (B (F, N), 5, d the bar resolution of N over F. We define

,

6),

is called

B'(M, F,N) = M ® B'(F, N) =MOkr Olc • • • Ok

r okN,

n times

with internal differential, dm 1 + (-1)d eg d., and external differential, 1 Or b. The next result follows from the definitions.

245

7.1. Differential homological algebra

Corollary 7.9. (1) H(total(B*(M, F, N)), total D) = Torr (M, N). (2) If F is connected and simply-connected (that is,

r' =

{O}), then the

spectral sequence of Theorem 7.6 converges strongly.

PROOF: The first part of the corollary follows from Proposition 7.8 and the definition of Tor. To establish the second part, observe that a typical element 7[71 ' • • 7nia E B —n (M, r, N) has internal degree deg -y + deg a +

deg

> 2n.

Thus each B' (M, r, N) is at least (2n — 1)-connected and the resulting filtration on total(B*(M, r, N)) is bounded. By the proof of Theorem 2.6 and Definition 3.8, the spectral sequence converges strongly. 111 Let us consider a simple example that illustrates a calculation with the bar construction. Let r = A(x), the exterior algebra over k with deg x = and trivial differential. To compute Tormx) (k, k) consider B . (k, A (x), k). In homological degree —n, we find n

13 —n (k,

times

A(x) 0k A Ok • • • Ok A Ok k

A(x),k) k

A Ok • • •

Ok

A (n times).

Here A = coker 7/ is the vector space generated by x in dimension m and so B' (k, A(x),k) has dimension 1 over k generated by [x • • • x] of bidegree (—n, mn). Since x2 = 0, the formula for the external differential gives us that = 0 and so Tor A (x )(k, k) B . (k, A(x), k). Tor*A(x) ' * (k k) can be displayed as a bigraded module giving the E2-term of the spectral sequence of Theorem 7.6 in the diagram: —4m —3m —2m

—

•

-

4

-

3

-

2

-

1

in

246

7. The Eilenberg-Moore Spectral Sequence I

Suppose (r, d) is any connected differential graded algebra over k for which H(F, d) A(x). By the placement of the nontrivial entries of TorA(x)(k, k), no nontrivial differentials can arise in the spectral sequence and so we conclude Torr (k , k)

Tor A(x)(k, k).

In §7.2 we apply this calculation to compute H* (S21; k).

Torr (k, k) enjoys an additional useful piece of structure. Proposition 7.10. If l' is a connected, differential graded algebra over k, then B• (k, r, k) is a differential coalgebra and this induces a coalgebra structure on Torr(k, k) as well as on the spectral sequence with E2 Tormr)(k,k), converging to Torr (k, k) as a coalgebra.

PROOF: The coproduct, A: B —n(k,r,k)

er+s _n B' ( k,r,k ) 013 —s (k , r,k),

is defined on a typical element [-y1 • • • -yn ] by AG-Yi • • •

I -ND =

i _o

I

•••

I

[-ri-ki

I

•••

I -y. ,

where the empty bracket, [] = 1, is taken when j = 0 or n. It then follows, by explicit computations, that A commutes with the external and internal differentials on B • (k, r, k) and so we have a differential coalgebra. The mapping induced on homology by A provides the coalgebra structure on Torr (k, k). Observe that the comultiplication A respects the filtration of 13 • (k, r, k) that gives rise to the spectral sequence of Theorem 7.6, that is, A (F— n) c 0 F 8 . This follows because the filtration in this case simply counts the "number of bars." Thus the spectral sequence is a spectral sequence of coalgebras and this coalgebra structure converges to the coalgebra structure on Torr (k, k). This coproduct has been applied in the study of local rings to determine the structure of the Poincaré series associated to TorR(k, k) (see [As smus591). There is a great deal of further structure that can be imposed on the Eilenberg-Moore spectral sequence via the bar construction and more generally from the structure of Torr (M, N). Furthermore, these structures have useful topological interpretations that we will present in the later sections of the chapter. Before leaving the bar construction, we consider some further structure in the special case of 13 . (k, F, k).

7.1. Differential homological algebra

247

Lemma 7.11. Let F and A be two di ff erential graded algebras over k. Then, as coalgeb ras, Torr (k, k)

TorA(k, k)

TorroA(k, k).

PROOF: We exploit another feature of the bar construction—it can be viewed as a simplicial object (§4.2). Let 13(F) = F (2'n and endow 13. (r) with face and degeneracy maps

I

ai(Vri

-•-

I ND

=

( -1 ) n [Y 1

si([-ril . • •

i=0 0
{ [-Y21.••17.1, (- 1 ) 1 [=y1

I -NJ = [-yi I

-•

H • •

I -Y.' I 1 I -ri+i I

-•

I

NJ,

0

< j < n.

The complex (13., d) with differential d = E i (-1)iai is called the unnormalized bar construction and it has the same homology as the bar construction (see [Mac Lane70D, that is, H (B . (F), d) Torr (k, k). The simplicial nature of the unnormalized bar construction suggests the key idea of the proof. Let B. (r) >< B. (A) denote the product of simplicial sets (recall that means ([3.(F) x (A)),„ = tri (r) x tri (A)). There is a natural identification

t.(r)

x 13.(A)

t.(r 0 A)

given by identifying [-yi I • • I • • • An] with [710 Ai • • • leY7i0A„] x and this is an isomorphism of coalgebras. To prove the lemma we compare 13. (r) 0 t. (A) and 13. (r) x t. (A) by taking the homomorphism at the heart of the Eilenberg-Zilber theorem (Theorem 4.36): There is a chain equivalence of coalgebras

t.(r)

(A) —> EZ

(F) x

(A)

t.(r 0 A)

given by the shuffle map

Ez([ y

,

• • eyp] 0

=

[At • • • AO

E( - 1 ) 47) [Ca(1) Ca(2) • • • ca(p+q)] (p, q)-shuffles,

cr

where c() = -y,(i) ® 1 if 1 < cr(i) < p and 10 A,( i)_p ifp+ 1 < c - (i) < p±q. The sign is determined by s(0- ) = E (deg ci ± 1) (deg ej-kp ± 1) summed over all pairs (i, j +p) with cr(i) > o- (p +j) (this is the sign convention that reflects a change of sign when elements are permuted past each other, according to their total degrees). The classical argument of [Eilenberg-Mac Lane531 completes the proof.

248

7. The Eilenberg-Moore Spectral Sequence I

The bar construction B. (k,r, k) is a functor from the category of differential graded algebras over k to the category of differential graded coalgebras. A differential graded algebra morphism 0: r A induces a mapping of coalgebras Toro (I, 1): Torr (k, k) —> TorA (k, k). If (r, d) is a graded-commutative differential graded algebra, then the multiplication is a morphism of differential graded algebras. Putting the induced mapping B' (k , F® r, k) —> B . (k, r, k) together with Lemma 7.11 we get the following result. Corollary 7.12. If (F, d) is a graded-commutative, differential graded algebra over k, then B . (k, r, k) is a differential graded Hopf algebra and Torr (k, k) is a commutative, cocommutative Hopf algebra with product induced by the shuffle product followed by multiplication on F. Furthermore, the spectral sequence converging to Torr (k, k) with E2-term Tormr ) (k, k) is a spectral sequence of Hopf algebras.

Finally, we record an algebraic consequence of the presence of a Hopf algebra structure on a spectral sequence that is most handy in topological applications. (See, especially, Chapter 10.) Lemma 7.13. If {E r , dr; r = 2, 3, ... } is a spectral sequence of Hopf algebras, then, for each r, in the lowest degree on which the differential dr is nontrivial, it is defined on an indecomposable element and has as value a primitive element.

PROOF: Since dr is a derivation with respect to the multiplication it follows from the Leibniz formula that in the least degree in which it is nontrivial dr is defined on an indecomposable element. Suppose that element is a. Then A(c17. (a)) = dr (A(a)) = = dr (a)

(a01+10a+

1 + 10 dr (a) +

cei 0a11)

i (d, (di ) di'

dr (aiii )) .

Since deg di < deg a and deg di' < deg a, the assumption of least degree implies dr (di ) = 0 = 4(4) and so (cir (a)) = dr (a) 0 1 + 1 0 dr(a). 7.2 Bringing

in the topology

We return to the basic problem of this chapter and the fibre square E1

E 7C

X

>B

249

7.2. Bringing in the topology

with 77 a fibration with fibre F and 7r' the fibration induced by 7r over f. Let C*( ) = C*( ; k) denote the cochain functor with coefficients in a field k. Applying C*( ) to the fibre square yields C*(B)-modules C* (X) and C* (E) via f* and 7r* and the cup-product. To make our "second guess" at H* (E f ; k) we compare Torc,( B)(C* (X), C* (E)) with C* (Bf) or H* (E f ; k). Define the mapping c: C* (X) øk C* (E) C* (E f) by the composite a: C* (X) Øk C* (E)

(7rTei*

> C* (E f)

C* (E f)

C* (E f).

In the definition of the tensor product over 1', we also have the homomorphism = (1 — f*) 1 — 1 (7r* — 1) : C* (X) Øk C* (B)

ek C* (E) —> C* (X ) k C* (E).

We show that a o = 0: a((x

e— x

f* (b))

= [(7r/ ) * (x = [(7ri ) * (x)

(77*(b)

f * (b))

e))

(f) * (01

(f ° 7rf ) * ( b)

[(7r/ )(x)

(f) * ( 7r* (b)

e)]

(f) * ( e)]

[( 70 * (x) — ( 7r ° f) * ( 1))

(f) * (e)].

Since 7rof=foir', we have o o = 0 and so a induces a mapping o : [C* (X) Øk C* (E) / ,m = c*(x) 6Dc* (B) c* ( E)

c* (Ef) .

Suppose Q* —>C* (X) —> Ois a proper projective resolution of C* (X) as a right C*(B)-module. There is a homomorphism E: total(Q) —> C*(X) given on total"' (Q*) by E on (Q°) 71 and zero on (Q —i )n+ i when i > O. Consider the composite 9 as in the diagram total(W) Oc* (B) C* (E)

C* (X)

c . ( B ) C* (E)

a

C*(Ef).

The relationship between the topology of the pullback diagram and the resulting algebra of cochains is described in a remarkable theorem of MilenbergMoore661.

250

7. The Eilenberg-Moore Spectral Sequence I

Theorem 7.14 (the second Eilenberg-Moore theorem). Suppose 7: E —> B is a fibration with connected fibre F for which the system of local coeffi cients induced by the fibre on B is simple. Suppose f: X —> B is a continuous mapping and 7r': Ef —> X is the pullback fibration of 7t over f. Then the mapping 0 — d o (s 0 1) is a homology isomorphism, that is, : total(W) Oc*(B) C* (E) —> C'(Ef)

induces an isomorphism on homology, : Torc* ( B ) (C* (X), C* (E))

H* (E f; k).

PROOF: The strategy of proof is to (a) filter the domain and codomain of 0 conveniently, (b) obtain spectral sequences that are isomorphic at E2 via 6 and (c) apply Theorem 3.5 to secure the desired isomorphism. Since C*(Ef) and C* (E) are each algebras of cochains of the total space of a fibration, they can be filtered as in the Leray-Serre spectral sequence (§5.1). Explicitly, we write Fq C*(Ef) = ker (i* : C* (Ef ) —> C* (Jr ))

where Jq-1 is the subspace of Ef that lies over the (q — *skeleton of X. f " If we filter C* (E) similarly over the skeleta of B. and filter C* (B) by degree, Fq C* (B) = ei, q ci(B), then we obtain a tensor product filtration on Fr (C"(B) 0 k C* (E)) =

i_o Fi C* (B) 0 k Fr— iC* (E).

The cochains in Fg. C* (B) are supported on skeleta of dimension at least q and so 7r * (Fq C* (B)) C Fq C* (E). Also, the cup product on C* (E) respects the filtration by inverse images of skeleta. Thus, the C* (B)-module structure on C* (E) is filtration-preserving: * Fq (C* (B) 0 k C* (E)) ®

Fi C* (E)

ok

C* (E)

FqC* (E).

i =0

Filter total(Q*) by total degree, that is, Fq (total(Q* )) = el i±i>q ( ) 1 ; this respects the C*(B)-module structure. Filter total(W) oc*( B) C* (E) as a tensor product Fq(total(V) 0c , (3) C * (E)) =

j=q Fi total(W) Oc-(B) F C* (E).

Such a tensor product on the right is possible because both filtrations preserve the C"(B)-module structure. Observe that the map 6 is filtration-preserving: : Fq(total(W) ®C(B) C* (E)) 0(f) *

€01

(C* (X) ®c(B) C*(E))

F q (C * (Ef (Sk C * (Ef))

Fq(C*(Ef))•

251

7.2. Bringing in the topology

The mapping E 0 1 is filtration-preserving, since it only sees the total degree and — is already known to be filtration-preserving; (7')* and (f)* preserve the filtration as geometric maps that preserve skeleta. Thus 6 induces a map of spectral sequences by Theorem 3.5. Recall from the construction of the Leray-Serre spectral sequence and the assumptions of the theorem, that the filtration on C* (Ef) has El -term, C* (X; H* (F; k)) and because k is a field and the system of local coefficients is simple, E2-term given by H* (X; k) k H* (F; k). For the filtered differential graded module, total(Q* ) Oc.( B) (E), we have the E0-term, Fq/Fq+1 = total(Q* ) O c .( B )FIC* (E) i+j-q

e

Ft total(W) Oc.(B) Fm C* (E)•

l+m=q+1

The filtration on total(Q) is given by total degree and ce + 6 raises total degree. It follows that do, the zeroth differential in the Leray-Serre spectral sequence for E, is zero on the total(Q) part. Since the filtration is decreasing, we compute that Fq Fg+1 [total(Q*)] EX* (E) and do = 1Ø d0. Thus E7'* = H(Fq/Fq+1 ) [total(W)]" 0c. (B) C* (B;7-t* (F; k)). Since the local coefficient system induced by the fibre is simple, we can write ET * [tOtal(Q. )]q 0C*(B) OkC * (B) Ok H * (F; k)

[total(Q • )] q ®k H* (F; k)

with dl given by (d• + 6) ® 1. To compute H(total(Q*), d• + s), recall from the proof of Theorem 2.20 that, if we view total (Q*) as a double complex, then there are two filtrations, one of which yields a spectral sequence that collapses to a column at the E2-term. This gives H(total(Q . ), di +6) = H(C* (X), dx) = H* (X; k).

Therefore, the E9-term derived from this filtration on total(Q*) o c .( B ) C* (E) is isomorphic to H* (X ; k) ek H* (F; k). To complete the proof of the theorem, it suffices to show that 6 induces an isomorphism of E2-terms. First, at the El -term, the induced homomorphism E10 total(Q) øk II * (F; k) —> C* (X) Ok H* (-F; k),

is induced by (f)* on the H* (F; k) factor and so, by the construction of a pullback, is an isomorphism. Furthermore, the mapping E1 6 on the total(Q* ) factor is induced by the map E, which is the mapping in the proof of Theorem 2.20 (the Kiinneth spectral sequence) that yields H(total(Q*), d• + 6) H (C* (X), dx). Thus. E20 is an isomorphism. By Theorems 3.5 and 5.2 (the Leray-Serre spectral sequence) we have H(total(Q*) o c .( B ) C* (E)) = —

Torc.( B)(C* (X), C* (E))

and Torc . (B) (C' (X), C* (E)) —> e*

(E f ; k).0

252

7. The Eilenberg-Moore Spectral Sequence I

The interested reader may want to rework the proof with coeffi cients in a ring R. We remark that the entire algebraic setting and topological interpretation (for pushouts) can be developed dually in homology (Milenberg-Moore66]); this is described later in the chapter. Suppose B is simply-connected. By normalizing the cochains on B, we can take C* (B) to be simply-connected as a differential graded algebra. Combining the first and second Eilenberg-Moore theorems with Corollary 7.9 (2), we obtain the solution to the problem posed in the introduction to the chapter.

Theorem 7.15. Suppose B is simply-connected, E B, a fibration with connected fibre F, and f: X —> B, a continuous mapping. Let 71-': Ef > X be the induced fibration over f given as a pullback. Then there is a spectral sequence, lying in the second quadrant, with —

E2

TOrH*(B;k)(H*

(X; k), H* (E; k))

and converging strongly to H* (E f; k). An immediate corollary to Theorem 7.15 is a spectral sequence with E2 term TorH , (x,k) (k, k) converging strongly to H(ÛX;k) when X is simplyconnected. More generally, the cohomology of the fibre of a fibration is recoverable from the base and total space.

Corollary 7.16. If7r: E B is afibration with base space B simply-connected and fibre F connected, then there is a spectral sequence with E9-term given by TorR*(B;k)(k, H* (E; k)) and converging strongly to H* (F; k).

With this corollary, we can apply the calculation of TorA(x ) (k, k) in §7.1 to a topological computation. Consider the fibre square Q82n+ 1

)

ps2n+1 2,

s 2n +1 .

By Corollary 7.16, H* (C2s2n-ki;, ■) is the target of the Eilenberg-Moore spectral sequence with E2-term E2i '3 Tori/e(x2 i) (k,k), where deg x2n+1 = 2n + 1. Our calculation shows that this spectral sequence collapses and we conclude the classical result—as vector spaces, B-to-2s2n+1 ; k)

totalt (Tor*4, 2a+i) (k, k))

if t = 2nr for r > 0, {0 } , otherwise.

k

In fact, we can do a little more here. Since S2S2n+ 1 is an H-space, we can wonder about the Hopf algebra structure on H*(Qs2n+1; k) . Since

253

7.2. Bringing in the topology

B -* (Qs2n+i; )

TorA(x , n+ ,)(k, k), there is a product structure already at

the level of the bar resolution. Define -y,([x]) = [x • • • x]

(m

times),

where [x] has bidegree (-1, 2m + 1) or total degree 2m. The shuffle product on these generators is given by ([x]) *-y,([x]) =

(-1)

p+q ([x]).

(p,o- shuffles 0 Since 2n + 1

is odd, s(a) is even for all (p, q)-shuffles and so -YP([x])

-Yq([xl) =

q)

where (P ± q is the binomial coefficient;

-YP+q([x]),

is the number of (p, q)shuffles. The reader can find this structure in the definition found in Exercise 1.7 of a divided power algebra on an element [x] of even degree 2n. The comultiplication resulting from the bar construction is described by (P ; 41

P

9 A(eYs([x]))

=

=0

( [x] )

-rs-, x]), ([

where -yo ( [x] ) = 1. Thus, TorA( x )(k, k) r([x]), the divided power algebra on [x], as a Hopf algebra over k. It is tempting to conclude that we have determined the Hopf algebra structure on H*(0S2n+ 1 ; k). However, to do that we need to assume that Torc . (s2—pi ;k) (k,k) is isomorphic as a Hopf algebra to TorH .( s2,+1 ;k)(k, k), and that the Hopf algebra structure coincides with the Hopf algebra structure on H* (0s2n+1 ; k). With a bit more effort, we will be able to conclude this. However, observe that C*(5,2,741; k) is not a graded-commutative differential graded algebra like its homology! This subtlety needs to be considered next. Multiplicative matters Our goal in this section is to prove that, in fact, the Eilenberg-Moore spectral sequence carries information on the cup-product structure of H* (E g; k). Proposition 7.17. On Torc9( B) (C*(X), C* (E)), there is a natural algebra structure and furthermore, : Tor ( B )(C* (X), C* (E)) —> H* (E f; k) is an isomorphism of algebras.

254

7. The Eilenberg-Moore Spectral Sequence I

To prove the proposition, we begin with a natural product structure on the Tor r (—, —) functor of the form 0k Torr (M2, N2) —> 1' Torrer ( ek M2, N1 Ok N2).

Torr

Let Ni and N2 be left r-modules and let

•• •

p-i Q -i

Q -i+i

—> • • • •• •

p° Q°

N2 —> 0

be left proper projective resolutions of Ni and N2 over L. Let

=e

m+n-i

P—m Ok (2 -n ;

the Kiinneth theorem implies that • • • —> R — i

R — i+1 —> • • • —> R ° —> Ni 0 N9 —> 0

is a left proper projective resolution of by the composite rereArioN2

1ØTØ1

Ni e N2

over F er that acts on Ni 0 N2

>reN1ereN2

N,

N2-

Let mereN 11/1- 0N be the mapping . (m0-y0n) = myOn m® eyn for M C DGModr and N c DGrMod. Recall that M 0r N = coker If Mi and M2 are right r-modules, then ini( 0 C NO 0 (M2 N2) is carried by the interchange 1 T 0 1 to the image of —

(mi e m2) ® (r e r) ® (N, e N2) f

loTei

ph ®FØ M2) e (N1 e r e N2) 10107, 0101

I

ph ® r ® No ® (m2 ® r ® N2 )

) (Mi

Ni

)

( M2 0 N2)

and so the interchange induces a homomorphism, for all m and n, (Mi

Ør Pm) 0 (M2 Or

Q -n) —> (M1 0 M2) Orer (Pm Ø Q — n ) •

This homomorphism induces the pairing TorP(Mi,

Ok Toq(/14-2, /\72) —> ToriV(Mi 0k M2, N1 Øk N2).

7.2. Bringing

255

in the topology

Recall from §4.4 that the coproduct on the chains of a space is the composition: A AW

„

C* (X

C(X)

X X)

—> C(X) C * (X).

This factorization, however, is not one of coalgebra mappings. In particular, AW is not a coalgebra mapping. The inverse of AW up to homotopy is the EilenbergZilber map EZ, based on shuffles; EZ is a mapping of coalgebras. Thus we get an honest coproduct structure on homology but only up to homotopy at the chain level. In cohomology, to factor the cup-product in terms of algebra mappings we meet a new problem: (C, (X) 0 C,(X)) dual is rarely C* (X) 0 C* (X)—it is a completed tensor product. We can, however, consider the maps C* (X) 0 C* (X) —>

EZ* (C* (X) 0 C* (X)) dual <_ C* (X

A*

X X)

—> C* (X),

where i denotes the inclusion into the completion. Every mapping here is a differential graded algebra mapping and on homology we can fit the induced homomorphisms into a product because EZ is a homology equivalence. Its inverse on homology is also an algebra mapping. To prove the proposition, we consider the diagram in which A = C* (B) TorA(C * (X),C * (E)) øk TorA(C * (X),C(E))

TorAeA(C * (X) Ok C* (X),

"e* 11* (E f) Ok

(-0 Ok (-0)

Tor,

TOr(c(B)0C(B))*((C(X)

H* (J)

EZ

C* (X)) * , (C* (E)®C* (E)) * )

TorEz ,(Ez* ,Ez*)I

Tor c (B x rt)(C * (X x X), C"` (E E)) Tors

Torc.(B)(C * (X), C* (E))

) H* (E f x E IA*

H* (E f).

We leave it to the reader to check that the Eilenberg-Zilber maps, EZ, commute with the actions of C* (B) and C* (B x B) on the appropriate modules, so they induce the vertical isomorphism as shown; since they are homology isomorphisms, Corollary 7.7 implies that TorEz* (EZ*, EZ*) is an isomorphism. We claim that the entire diagram commutes. The bottom square commutes by the naturality of the homomorphism 6 and the geometric mappings of

256

7.

The Eilenberg-Moore Spectral Sequence I

librations Ef A

FxF—Ef

X

and

A

Ef -, XxX

FxF—,ExE-13>
The top square commutes because the morphisms constructed to carry the mappings cover the composite A* o (EZ*) -1 oi*, which is the cup-product. Since the composite on the left side is the product induced by A on Tor and the product on the right is the cup-product, we have proved Proposition 7.17. In the proof of Proposition 7.17, the reader can check that all maps in sight preserve the filtration leading to the Eilenberg-Moore spectral sequence. This implies the following useful result. Corollary 7.18. The Eilenberg-Moore spectral sequence is a spectral sequence of algebras, converging to its target as an algebra.

In the special case of the path-loop libration, our factorization of the cupproduct into differential graded algebra morphisms, albeit dizzying in direction, induces a Hopf algebra structure on Tor because the crucial (backward pointing) morphism is an equivalence. Corollary 7.19. The Eilenberg-Moore spectral sequence associated to the pathTorH. (x;k)(k,k) loop fibration on X, a simply-connected space, with E2 is a spectral sequence of Hopf algebras, converging to H* (S2X;k) as a Hopf algebra.

In §7.1 we computed Toro. (k, k) F ([x] ) for any differential graded algebra (A*, 0) with H (A* , 0) A(x) and deg x odd. Corollary 7.19 implies that F ( [x]) E° (H* (S2S 2n+1 ; k)) as Hopf algebras. To obtain the Hopf algebra H* (S 2 '; k) exactly, we need to settle the extension problems. As a vector space, there are no problems as r ( [x] ) is concentrated in even degrees. As an algebra the extension problem depends on the characteristic of k. If char k = 0, then r ([x]) is isomorphic to a polynomial algebra on one generator and there are no vanishing products. As in Example 1.K, we have established the product structure completely. If the characteristic of k is finite, then we must turn to another argument. Here the structure of the Leray-Serre spectral sequence for the path-loop fibration settles the extension problems. To wit, the issue at hand is the vanishing of pth powers of the class [x]. They vanish in the structure of a divided power algebra (Proposition 7.26). In the cohomology of S282n+ 1 , the appearance of a pth power implies a class on which the first (and only) nontrivial differential in the Leray-Serre spectral sequence vanishes. This

7.3. The Koszul complex

257

leaves a nontrivial class in H+( p5,211+1 ; K) and gives a contradiction. Thus the product structure on H* (0s2n+1; k) coincides with the product structure of r([x] ) and we have recovered H* (S2S 2n ; k) as a Hopf algebra. The cup-product on cochains fails to be a differential graded algebra mapping. However, special circumstances (an explicit and nice homotopy inverse) allowed us to produce an algebra structure on the spectral sequence. 1f (A*, dA) and (B*, dB) are differential graded algebras, we can ask more generally for conditions on a k-module morphism (A*, dA) —> (Bk, dB) to define a morphism on Tor, even for the case of Tor(A.,d,)(k, k) —> Tor(B.,d,)(k, k). This question with A* = C* (X; k) C* (X ; k) and B* = C* (X ; k) includes the case of the cup-product. We discuss this question in the next chapter. 7.3 The Koszul complex

With the bar construction and Corollary 7.19 the Eilenberg-Moore spectral sequence leads to the computation of H* (0,92n+1; k) as a Hopf algebra. The E2-term in this case is rather simple and the spectral sequence collapses. In contrast, the same computation with the Leray-Serre spectral sequence (as outlined in Example 1.H) involves a flurry of nontrivial differentials, some of which settle the extension problems for the E -term of the Eilenberg-Moore spectral sequence. Lest the reader come to believe that the Eilenberg-Moore spectral sequence is a more facile method for computation, we describe a symmetric situation where the Leray-Serre spectral sequence associated to a fibration is quite simple while the Eilenberg-Moore spectral sequence requires a panoply of differentials. Consider the Hopf fibration, S 3 S 7 -which is associated to the quaternionic multiplication. We can compute H* (S 3 ; k) as the cohomology of the fibre of v by applying the Eilenberg-Moore spectral sequence. Here the E2-term is Tor (S4 ; k)(k, H* (.5'7 ; k)) which we compute using the bar construction. In particular, as a vector space, 13 –n (k, H* (S4 ; k), H* (S 7 ; k)) is generated by n

times

[X4 • • • X4]

n

and

[ 3 4 • • • X4] Y7-

On each generator, the exterior differential, TorH*(s4 ;k) (k , H* (S 7 ; k))

times

6,

is zero and so

B • (k, H* (S 4 ; k), H* (S 7 ; k)).

By a dimension argument, the only possibly nontrivial differential is d2 . If d2 ( [x4 x4]) = [ ]y7, then the multiplicative structure implies the rest of the nontrivial differentials needed to obtain a finite-dimensional E -term. Thus

258

7. The Eilenberg-Moore Spectral Sequence I

[x4]

remains to generate H* (S3 ; k) in total degree 3 = 4 — I. x4[y7 y7

[x4 I x4 I [x4 I X4]

612

1 Y7 [x4]

-3

-2

-1

This computation is not at all efficient as a way to determine the cohomolof the fibre of v. It does reveal, however, some of the algebraic topology associated to the Hopf fibration: In Chapter 8, the nature of the differentials in the Eilenberg-Moore spectral sequence is discussed and the interpretation of differentials in terms of higher cohomology operations may be applied in this case. The reader can contrast this computation with the discussion of the Leray-Serre spectral sequence for fibrations of spheres by spheres after Example 5.B. In any application of the Eilenberg-Moore spectral sequence, it is necessary to compute TorH(r)(H(M), H(N)) in order to begin with the E2-term. In the examples we have considered so far, the bar construction has been manageable as a result of the sparseness of the algebraic input. For algebras with more than one generator and some relations, the bar construction can become quite large and complicated. One of the features of homological algebra is the invariance of the derived functors with regard to the choice of resolution and so the construction of smaller and more manageable resolutions is important. Though this cannot be achieved in general, there is a considerable reduction possible when H(F) is a free graded commutative algebra over k. Let J denote a graded set. The free graded commutative algebra, S(J), generated by J is determined by a universal property: The set J includes in S(J) as a generating set for the algebra. If A is any graded commutative algebra over k and J —> A any mapping of sets, then there is a unique mapping of algebras S (J) A extending J —> A. We may write J = jeven u jodd where jeven { XE J1 deg x is even} an d jodd _ jeven (If char k = 2, then let Jevell = J). If J is a singleton set J = {x}, then S({x}) k[x] when x has even degree or char k = 2, and S ({x}) `=' A(x) when x has odd degree. If J has finitely many elements in each degree, then an application of the universal property shows S(J) k [ jeven] ok A( jodd), that is, the tensor product of the polynomial algebra on the elements in J" and the exterior algebra on the elements in Pdd. ogy

259

7.3. The Koszul complex

Definition 7.20. Let J be a positively graded set (that is, Jr = 0, for r < 0) which is finite in each degree. The Koszul complex associated to S(J) is the differential graded algebra 1C(J) = (G(.3-1 J) 0k S(J), dic),

where s — 1 J is the desuspension of J, given by the graded set (.3' and G( 5 -1 j) A( 3 -1 jeven) ok r(

= Jm+1

j odd ).

The differential di( is given on generators by dic(1 x) = 0, for x S(J), dic(s -1 x 0 1) = 1 O x, for x x, for x G rdd,

x) 01) =

and extended as a derivation. Proposition 7.21. For J a positively graded set that is finite in each degree, the homology of the associated Koszul complex is trivial, that is, {0 } .

H (1C(J), dic)

PROOF: If J = {x} is a singleton set and x has even degree, then the argument given in Example 1.H applies to prove this case. If J = {x} and x has odd degree, then observe that l'(s'x) B (k, A(x), k) via the mapping, •-y.(s - ix)

[x

I

•••

I

x]

(in times).

It follows that 1C({x } ) total B' (A(x), A(x)), which is acyclic. To complete the proof for an arbitrary generating set J, filter J by degree and apply the Kiinneth theorem and induction to the successive filtrations. In order to construct a resolution from the Koszul complex, we introduce a bigrading: Because we can write

lc ( j )

(A( s -1 jeven)

g k r(s _i jodd )) 01, (k[ jeven] 0k A( jodd)) ,

we assign the bigradings as follows: If x E J, bideg (1 o x) = (0, deg x) and bideg(s'x 1) = (-1, deg x) and, if y E J°dd , bideg(-y,, (s —l y) 1) = (—n, n deg y). All other elements in 1C(J) are bigraded as in the tensor product of bigraded algebras. Taking G(s' J) to be bigraded according to this scheme, we observe that 1C — P, * (J) = G— P, * (.3 -1 J) Ok S(J), that is, IC — P, * (J) is an extended S(J)-module. Furthermore, the differential on the Koszul complex is defined so that dk : 1C — P'*(J) —> /C—P+1'*(J).

260

7. The Eilenberg-Moore Spectral Sequence I

Corollary 7.22. The sequence, dic • • • —>

dic P'* (J) —> • • —>

l '*(J)

dic

S(J)

k

0,

is a resolution of k as a right S(J)-module, called the Koszul resolution.

We apply the resolution immediately to compute Tors (I) (k, k). Following Definition 7.5, Tor so (k,k) = H (K (J) 0 s( j) k,dx, 01) = H(G(8-1 J) 0 S(J)®S(J) (0k,

0s(j) 1)

= H(G(8-1 40) = G(s-1 J).

More generally, suppose L is a left S(J)-module. Corollary 7.23. For L a left S(J)-module, Tors ( j )(k,

H(G(s - 1 J) Ok L,d1),

where the complex G(8 -I - J) L has the differential, 4, given by 4(101) = 0, c/L(s -l x e = 1® xl, for x E J, 4(7ri (3 -1 x) e = 7_i(s 1 x)® xl,f0rx E Jodd

If L is a graded commutative algebra over S(J), then Tor s ( j) (k, L) is isomorphic to H(G(s -1 J) o L, 4) as bigraded algebras over k.

The result follows from Corollary 7.22 and the definition of Tor: Tor s ( J)(k, L) H(G(s -1 J) øk S(d) s(j) L, d 0s( j) 1) H(G(3 -1 J) Ok L.4).

To recover the algebra structure on Tor s( j) (k, L) check that the following diagram of graded algebras commutes: G(8 -1 J)0LoG(s -1 J)0L—L0L

G(s -1 J) L

L.

We next apply this new tool to a familiar cohomology computation, Example 5.G. The complex Stiefel manifold, V/ (Ca ), of (-frames in Ca , is homeomorphic to U(n) /U(n -1), the quotient of the unitary groups, U(n -1) C U(n),

7.3. The Koszul complex

261

by the usual inclusion. We have already remarked that a homogeneous space GI H can be taken to be the fibre of the mapping Binc BH —> BC. Thus we have the fibre square BU(n — 1)

(cn)

Bine

BU (n) • The Eilenberg-Moore spectral sequence for this fibre square begins E2"

Tor *k i% ,

1

(k, k[c2, • • • ,

therefore, we can apply Corollary 7.23 to compute the E2-term. The action of k[c2 , , c2 ] on k[c2 , , c2(n _ i) ] may be described by writing the algebra k[c2, , c2(n _ i )] as the quotient k [c2... • , C2r]/(C2(n-/)+2, • • • • C2/1). Corollary 7.23 gives the complex A(xi, • • • , X2n-1) ODk

k[C2, • • • • C2(n-/)],

which we can rewrite as A(X2(n-i)±1,

• X2n-1) Ok A(Xi,

X2(11-0-1) Ok k [C2, • • •

The explicit differential on the complex is fashioned out of the k[c2 , . . . , c2n]action on k[c2, . . . ,c2 (n _o]. This complex splits as the acyclic Koszul complex (A(xi , • • • , X2(n-/)-1)øk k[C2, • • • • C2(11-0], dK;) for k [c2, • • • , C2(n-I)j, leav ing E2 A(X2(n-1)±1, • • • • X2i1-1). Because all of the algebra generators for E2' lie in E2-1 ', the spectral sequence collapses and A(X2(n—/)+1,

• • • • X2r1-1)•

Since this is a free graded commutative algebra, there are no extension problems and so we conclude, as in Example 5.G, H'(Vi (Cn ); k)

A(x2(ri—o-ki,

, x2,2-1) •

The Koszul resolution plays an important role in the study of the cohomology of homogeneous spaces to be considered more deeply in Chapter 8. The free graded commutative algebra S (J) when J is locally finite enjoys some further structure—S (J) is a graded commutative Hopf algebra. The Hopf-Borel structure theorem (Theorem 6.36) allows us to write a locally finite graded commutative Hopf algebra H over a perfect field as a tensor product of monogenic Hopf algebras. To compute Torn- (k, k) in this case we can use the splitting of H into monogenic factors (Lemma 7.11). The Koszul complex handles the free graded commutative pieces leaving the case of a truncated polynomial algebra factor over a field of characteristic p> O. This case admits a similar resolution described by [Tate571 in the ungraded case and extended to the graded case by klozefiak721.

262

7. The Eilenberg-Moore Spectral Sequence I

Proposition 7.24. Suppose that k is a field of characteristic p > O. Then Tor k[x]/(x 2, (k k) A(a 2 - 1 ) Or(b2nps —2), where a2,„ 1 corresponds to ,

)

[x27,] and b22 corresponds to [xF2:» x27,] in the bar construction.

The element b2n ps _2 is called the transpotence element as it plays the role of the element (Tx)P s —1 x in the Leray-Serre spectral sequence (see §6.2). PROOF: To carry out the computation we proceed naively. We can form (A(a2 n _i) 0 k[x 2 7i ] I (47,), d) with differential given by d(a2n _i) = x2n , d(x2, ) = 0 and extended as a derivation. This does not yield a resolution as the homology of this complex is nonzero. In particular, there is a nonzero homology class given by a 0 xP' — and the homology is isomorphic to an exterior algebra generated by this class. Next form the differential graded algebra (b2rip s _2) 0 A(a2

1) 0

where d' = d on the subalgebra where d is defined, and d' (b) = a xP' . The extension of the differential to the other generators of the divided power algebra is given by (-ym (b)) = -ym _ 1 (b) ®a ® x'.

The proposition follows by showing that this complex is acyclic. If we bigrade the complex according to the rule bideg(x) = (0, 2n), bideg(a) = (-1, 2n) and bideg(b) = (-2, 2np'), then filtering by homological degree leads to a spectral sequence that separates out the r (b) factor from A (a) k [x2.] I in the first term and looks like r(b) ® A(a ® xif — ') in the next term. The argument for the Koszul complex proves acyclicity and the bigrading provides the resolution.

(xl2n 1)

When (X, ,a) is a locally finite H-space, H* (X; k) is a graded commutative algebra and TorH.(x;k)(k k) is the E2-term of the Eilenberg-Moore spectral sequence that computes H* (QX; k). With such complete control of the E2-term in this case, lGitler621 and lClark651 proved the following result. Hopf

Theorem 7.25. Let (X, ,a) be a simply-connected H-space that is of the homotopy type of a finite complex. Then, in the Eilenberg-Moore spectral sequence converging to H* (SU; for p a prime, the only possible nontrivial differentials dr satisfy r = p8 — 1 or r = 2ps — 1. )

PROOF: From the assumptions we know that H*(X;7,) is isomorphic as an algebra to a tensor product of the following form II* (X;F p)

A(xl , • • • ,x m )® ( 72[M]l (Yr )) 0 • • •

p[Yn]l (Y 71):' )) •

7.3. The Koszul complex

263

It follows that TorH. (X;Fp )(7p, 719) = E2 takes the form F(a i ,

, arn )

A(bi ,

r(ci ,

, cri )

where the ai have bidegree (-1, lxiD, the bi have bidegree (-1,1 y ) and the ci have bidegree (-2, y3 ) . By Lemma 7.13 the lowest degree differential takes an indecomposable to a primitive. We first consider the primitives. According to the Hopf algebra structure on E2, the primitives are spanned over 7, by the set {a, bi , Since the bidegree of the differentials is (r, 1 r) and r > 2, the primitives are all permanent cycles. To determine the indecomposables of E2, we need to know a little more about divided power algebras over Fp. Since a divided power algebra is locally finite and commutative, it is described by the Hopf-Borel theorem as a tensor product of monogenic Hopf algebras. —

Lemma 7.26.

r u21 (

)

i=0

PROOF: Let u = u21. The Poincare series for FM is given by P(r(u), t) = 1 . This can be seen by identifying r(u) with 13*(Fp , A(su), Fp). The 1 — t21 Poincaré series for the tensor product of truncated polynomial algebras is given by 00 t21p z+1 1

EL

1_

= 1 — t 21

t2 P

Notice that the tensor product of truncated algebras has the property that every element has pth power zero. We note that this is also true in r (u): Given any generators, -yr (u) and 'y (u), the defining relation for 1'(u) is 'Yr (u)'Y

= (r +8 s )

+ ,(u).

By induction (-yr (u))P = (Pr) il9 r (u) and the binomial coefficient is zero mod p. r Since every pth power vanishes in F(u), there is a homomorphism of algebras 0: ® [-y, (u)]1((-4,i(u))P) (u) taking -, 4, (u) to itself. From i=o P P the equality of the Poincaré series it suffices to show that this homomorphism is an epimorphism. Recall Lucas's Lemma (ftlardy-Wright381): If A and B are positive integers and A = cto + alp +... +aupv and B = 0o + 01P+ • • • + Oupu are the base p expressions of A and B (in particular, 0 < ai , < p), then

() "

mod p.

264

7. The Eilenberg-Moore Spectral Sequence I

We show that we can express a typical generator 7A (u) in the divided power algebra in terms of the -yp,(u): Write A = cto + ct ip + • • • + av pv Claim: -yA(u) =

p (U)) a l • • • (eyp ,

C A(eyi(11)) a°

Mr' with CA

0 modp.

We prove the claim by induction on v. For v — 0, there are only the generators -y,(u) where 1 < j < p. We have the identity eyi (u) = in this range and so the induction is begun. Assuming the claim for sums out to pv — ', we write

A=

/i (-Yi (u))i

ao + alp + • • • + auPv

= (to +Ptat + • • • + oe vpv-1 ) = cto + pA i

The defining relation for the divided power algebra gives (cto +PA') 7A (a) = -rao(a)-xpA , (a)• cto

By Lucas's Lemma, (cq) + PA' ) ra 1 mod p and so we have A(U) =

Ca o (eyi (u)) a ° 72,A, (u).

Consider the mapping F: -yi (u) 1 > ypi (u) defined on f (u) . Though this is not a morphism in the category of graded vector spaces, it is nonetheless a multiplicative monomorphism by Lucas 's Lemma. By induction we can express A' (u) as in the claim. Apply this Frobenius mapping F on r(u) to obtain the expression for -yA (u) in the claim. It follows that the homomorphism 0 is an epimorphism and the lemma is proved. —

The decomposition in Lemma 7.26 implies that the indecomposables of Torip pc ; 7, ) (Tp , 7p) lie in bidegrees (—p 8 , (deg ai)p8 ) or (-2p8 , (deg ci)p8 ). These bidegrees have even total degree and so a differential cannot have an image among the primitives of even total degree. Thus a differential that is nontrivial must land in bidegree (-1, even). It follows that the first nontrivial differential in the spectral sequence must be defined on En where n = p8 — 1 Of

n = 2p8 — 1.

If we take such a differential and restrict our attention to its domain and image, we find a differential Hopf algebra of the form (r(u) A(V), d), where U = -yps (u21) and d(U) = V, a class among the odd degree primitives in Tor —l '*. The homology of this piece of the spectral sequence is given by

H(r(u)e, A(V),

r(u)/eyt (u) t >p3).

Though this homology produces a new primitive, it is in an even degree and so determines a permanent cycle. Furthermore, no new indecomposables are

7.4. The homology of quotient spaces of group actions

265

created. If we compare the initial differential Hopf algebra with the quotient by this differential sub-Hopf algebra, then we get a splitting of any En -term into differential Hopf algebras of this form and ones with trivial differentials. This proves the theorem. More can be said about the differentials that appear in Theorem 7.25. We postpone the discussion to Chapter 8 where we develop a general setting for describing differentials in the Eilenberg-Moore spectral sequence. Theorem 7.25 plays an important role in the study of torsion in H-spaces—one of the main topics of Chapter 10 (see the paper of IKane751). 7.4 The homology of quotient spaces of group actions

In this section we consider another situation where the differential homological algebra of §7.1 may be applied and the spectral sequence of Theorem 7.6 interpreted topologically. We assume that the reader is acquainted with the following notions: G is a topological group, and G acts on spaces X and Y. Denote these actions by ipx : X x C —> X (a right action) and 'y : C x Y Y (a left action). To such data, we associate a new space, X xG Y, the pushout of qpx x 1 and 1 x 1Py, as in the diagram XxGxY

.0x xi

X xY

X xY

X x G Y,

that is, X xG Y is the quotient of X x Y by the relation (xg, y) (x, gy) for all x E X, y E Y and g E G. Some examples of this situation are: (1) Suppose G acts on Y and Y/G is its orbit space. Then, if X = *, a point endowed with the trivial action of G, we recover Y/G as * x G Y. Observe that whenever the action of G on Y is free, we get a principal fibration, Y —> Y/C.

(2) An important example of the situation in (1) has been described in Chapter 6: When EC is an acyclic space on which G acts freely, then EG/G = BC is the classifying space for principal G-bundles and we have the fibration G EC BC. (3) Suppose G E —> X is a principal G-bundle and F is a right G-space. Then we can form the fibre bundle over X with fibre F and structure group G. This is the associated fibration F F XG E —> X = E /G.

This construction is discussed in §6.2. From the description of X xG Y, we pose the problem dual to the problem of the introduction to the chapter.

266

7. The Eilenberg-Moore Spectral Sequence I

Problem. Compute H* (X x G Y; k) from knowledge of H * (X; k), H * (Y ; k), H,.(G;k) and the actions of Con X and Y.

In order to place ourselves in the algebraic setup of §7.1, apply the singular chain functor, with coefficients in a field k, to the G-action maps. We write C* ( ) for C* ( ; k). Then G (G) is a differential graded algebra with the Pontryagin multiplication induced by the group multiplication on G. Further we have actions of C* (G), C(X) C(G)

C(X)

and C(C) C(Y)

C(Y)

induced by 'Ox * and 1Py* . Therefore, C* (X) and C(Y) become right and left modules over C* (G). In this setting we can construct a proper projective resolution of G (X) as a right C* (G)-module, Q. -# (X) -> O. Note that Q. can be presented as a first quadrant double complex since the differential on C* (X) has degree —1. From the pushout diagram, there is a mapping of G(G)-modules, : C* (X) 0c,(G) C(Y) —> C * (X x

G

Y).

We introduce the composite 0, as in the diagram total(Q*)

C. (G) C(X)

6011

C(X) ®C(G) C(Y)

a

C,(X xc Y).

The following theorem is due to 1Moore591. Theorem 7.27. Suppose G is a connected topological group, X is a right Gspace, and Y is the total space of a principal G-bundle where G acts on Y on the left. Then the mapping 8 is a homology isomorphism, that is, there is an isomorphism 0: Torc* (G) (C,,(X),C,,(Y)) —> H,,(X x

G

Y)•

The proof follows the same outline as the proof of Theorem 7.14 with homology substituted for cohomology. The source of the comparison with the Leray-Serre spectral sequence is the fibration induced by the G-map X —? X —> X >
Further details are given in the exercises.

7.4. The homology of quotient spaces of group actions

267

Observe that we have displayed the variance of Tor by writing C.(G) as a superscript. We follow the conventions described in Chapter 2—all the differentials in sight have degree —1 and so our resolutions are positively graded. In particular, Q. —> C(X) —> 0, the resolution, and Tor c* (G) (C. (X), C.(Y)) are bigraded with both integers of the bidegree nonnegative. We can filter total(Q.) (X)G. (G) c (Y) in the same fashion as in the proof of Theorem 7.6 and the same arguments (this time in the first quadrant) yield the following result. Theorem 7.28. Suppose G is a connected topological group, X is a right Gspace, and Y is the total space of a principal G-bundle where G acts on the left on Y. Then there is a first quadrant spectral sequence with E2 Tor11* (G ;k) (H.(X ; k), H.(Y k))

and converging strongly to H5 (X X G Y; k).

In order to compute Tor in this case, the bar construction, suitably regraded, provides a proper projective resolution, n times

(M, F, N) = M

r ® • • • ® r erN,

where r denotes the cokernel of the unit of r. In the special case of M = k = N, Proposition 7.10 still applies and B. (k, r, k) is a differential coalgebra with the usual comultiplication. In this case, the spectral sequence of Theorem 7.21 converges to the homology of a space that has a natural coalgebra structure induced by the diagonal mapping. Since this is a homological feature of the homology of a space, it is no surprise that our constructions determine it. Corollary 7.29. Let G be a connected topological group. Then the mapping O induces an isomorphism of coalgebras, O.: Tor c* (G) (k, k) H.(BG;k). Furthermore, there is a spectral sequence of coalgebras, with E2

Tor11* (G ;k) (k, k)

and converging to H.(BG;k) as a coalgebra.

SKETCH OF PROOF: Regard B. (k, C(C), k) as a coalgebra and filter it as in Theorem 7.14. This filtration respects the coalgebra structure and induces the desired coalgebra structure on Tor IL(G ; k) (k, k). Thus we have a spectral sequence of coalgebras, converging to Tor G * (G) (k, k) as a coalgebra. To show that Torc* (G) (k, k) is isomorphic to H .(B G; k) as a coalgebra, it suffices to compare B. (k, C5 (G), k) with C.(BG) via the classical description

268

7. The Eilenberg-Moore Spectral Sequence I

of the comultiplication on a space. The comparison is implicit in the paper of [Mi1nor561. It also follows from the simplicial construction of BC given by [Stasheff63] and [Milgram671. The spectral sequence of Corollary 7.29 has received considerable attention in the literature. It has been referred to as the bar spectral sequence, the Milnor spectral sequence, the homology Eilenberg-Moore spectral sequence, the Moore spectral sequence, and the Rothenberg-Steenrod spectral sequence. In each case, II (BC; k) is the target of the spectral sequence. The problem of computing this homology may be approached in two ways. The first is algebraic in nature, due to Eilenberg and Moore, as sketched in this section. The second approach is geometric and begins with Milnor's description of the space BC as the direct limit of quotients of iterated joins of the group G ([Milnor561). This construction provides a filtration of BC by join coordinates and thus a spectral sequence converging to I-I* (BG;k). [Milnor561 identified the E'-term of the spectral sequence as

4,1

(G; k) ® • • • 0 -Hi,(G;k).

He also gave an explicit expression for the first differential in terms of the iterated diagonal mapping. [Rothenberg-Steenrod651 analyzed Milnor's construction further and computed the E2 -term of the spectral sequence along with the further structure described in Corollary 7.29. In order to show that the algebraic and geometric constructions of these spectral sequences yield the same spectral sequence, a comparison at the level of chains is needed. This comparison was carried out by [Stasheff63] and [Milgram671 who introduced a different construction of BC that compares well with Manor's along with a filtration that leads to the same spectral sequence and carries enough structure to allow a comparison of chains. Their definition of BC is given in Chapter 6. An equivalent and elegant definition of BC as a geometric bar construction, BC = B(*,G,*),is given by [May751. The space B(*,G,*) is the geometric realization of the simplicial space, B.(*, G, *), where the space of n-simplices

is given by B n (*,G,*) =GxGx•••xC (n times), and the face and degeneracy maps are given by [g21

ai( [91, • • •

,gri]) =

...

[gi, • • •

{

, gn], ,

g

• gi+i, • • • ,

gab

[gl, • • • ,g,i_i],

if i = 0, if 1 < i < n, if j = n,

si([g i , • • • ,grt ] ) = [g i , • • • , gi, e, gi+i, • • • ,gn]•

An explicit chain equivalence B. (k , C * (G), k) C (B (* , G , *)) can be given. These comparisons imply the identification of the algebraic and geometric spectral sequences as the same.

7.4. The homology of quotient spaces of group actions

269

Recall that the functor Tor_ (k, k) respects tensor products of algebras. We apply this fact (Lemma 7.11) to give a different proof of Theorem 6.38 of [B ore1531: If G is a connected topological group with H.(G;k) , x r ) as an algebra over k, where deg x, is odd for all i, then

Theorem 7.30. A(xi,

H* (BG; k)

k[y"; ,

,

as algebras, with deg y — deg x i + 1.

PROOF (Noore591): As algebras we have the isomorphisms H(G;k) A(x 1 ,

, x r) A(x i ) 0 • • • 0 A(x r ).

Consider the E2 -term of the spectral sequence of Corollary 7.29, E2

TorH* (G;k) (k, k)

TorA( '' 'x ') (k, k)

By the same argument that computes H" (1S2 resolution yields

TorA( ') (k, k).

'; k) as a coalgebra, the bar

TorA( xi ) (k , k)F(y)

where (y3) is the divided power algebra on the generator Yi = [xi] with deg yi = deg xi +1 (recall the bar resolution is positively graded in homology). Thus, as coalgebras, E2 r(Yi, • • • 'Yr.) with all y, are of even total degree. Since every element of total(E2 ) has even degree and differentials decrease total degree by 1, the spectral sequence collapses and E° (H(BG; k)) (y ' , ,y r ). By an argument dual to the one given in Example 1.K for free graded commutative algebras, r(yi , , tir) plays the role of a free cocommutative coalgebra and so H.(BG; k) (y i , , yr) as a coalgebra. Since all the elements of H.,(BG; k) lie in even dimensions, H* (BG; k) H,,(BG; k) dual r(yi,

yr) dual ,

as algebras. We leave it to the reader to prove that, as an algebra, the dual of the coalgebra (y i , , tir) is the polynomial algebra k[yI, , yr,*] with 4 dual to y3 . This completes the proof.

7.

270

The Eilenberg-Moore Spectral Sequence I

Exercises 7.1. Prove that every proper projective differential graded module over r, a differential graded algebra over a ring R, is a direct summand of an extended module over F.

7.2. On Torr (M, N): a) Verify that the definition of Torr (M, N) given in Definition 7.5 does not depend on the choice of proper projective resolution of N. b) Verify that Torr (M,

N) is the homology of the complex total(Q) Or N with 0 1 ± (_1)deg 0 a proper projective dN for Q* - M

differential D resolution of M in DGModr.

6 )

-)

c) Verify that, given a short exact sequence in DGrMod,

0 —> N and

M

—> N2 —> N3 —>

0,

in DGModr, we get a long exact sequence of modules over

R

a a

d) Verify that, for a proper projective module all M and all i > O.

P over r, Tori, (M, P) = {0}

for

e) Verify that Torr (M, N) is a functor in each variable separately and determine what conditions are necessary to consider Tor as a functor in three variables.

(r, N),

7.3. Show that the bar construction, B

is a resolution when dealing with algebras and modules over any ring R. Show, however, that some further flatness conditions are needed in order to guarantee

H (B' (r, N), dint)

B*(H(r), H (N)).

7.4. On the bar construction. a) Verify that (5 6

=0

where

b) Verify that d' d resolution.

6 is the external differential

= 0

where d' is the internal differential for the bar

c) Verify that d' is F-linear and that d) Verify that d– n –l s

B' (r,N).

for the bar resolution.

6o

sd' = 0

where

± (I -71 +1 o = s

O.

is the contracting homotopy on

Exercises

271

7.5. In the Eilenberg-Moore spectral sequence to calculate H* (F; k) for F the fibre

of a fibration F

E

—

> B, show that =

H* (E; k)

—

> H* (F; k)).

7.6. Suppose that r : E —> B is a fibration and f : X —> B a continuous mapping. Show that the Eilenberg-Zilber map EZ: C* (B x B) > C* (B) commutes with the C* (B X B) and C* (B)-module structures on C* (E x E) and C* (E), respectively, as well as the module structures on C* (X X X) and C* (X), respectively. —

-

7.7. On the homology Eilenberg-Moore spectral sequence: a) If C is a differential graded coalgebra over a field k with counit E, then a comodule over C is a differential graded module A over k equipped with a comodule mapping A: A > A Ok C so that the following diagrams commute: —

A

A

)

Al A

A

C

A

A

1A01

C

liDA

A

AOCOC

A

C

106 > A

k

Given a fi bre square

Ef

E Ic

X

>B

verify that C, (X; k) and C, (E; k) are comodules over C, (B; k). b) Define the cotensor product

kB

AlIGB = ker (A

AA01-10)03

>A

e k c ok B)

when A is a right comodule over C and B, a left comodule. Verify AlicC = A. c) Give a definition for the notion of an injective comodule over C as dual to the idea of a projective module over an algebra. Define the idea of an injective resolution and then define

CotorC (A, when 0 —> A —> X0 —> X1 —> comodules over C.

= H (X.111c 13; d ® 1) X2 -> • • •

is an injective resolution of A by

d) State and prove the homology analogue for Cotor of the first and second Eilenberg-Moore theorems (Theorems 7.6 and 7.14). (The reader is directed to the classic paper of [Eilenberg Moore661 for complete details.) -

272

7. The Eilenberg-Moore Spectral Sequence I

7.8. Prove Corollary 7.18. 7.9. Prove Theorem 7.27. The following strategy is helpful: Begin by considering the principal C-bundle, C Y * x a Y, that is, prove the theorem first for X = *. The general case follows the proof of Theorem 7.14 and applying those ideas to the fibration

X X xG Y

> Y/G = * xGY 1

—

7.10. Use the bar construction to compute H* (QC P (n); k). Use a Koszul complex to do the same computation. 7.11. Prove that the divided power Hopf algebra F (y) on the generator y is the free cocommutative coalgebra on y. Prove that the dual of the coalgebra F(y) is the polynomial algebra k [V] where y* is the element dual to y. 7.12. Suppose there is a libration, F E B, with E acyclic and H* (B; k) a polynomial algebra on finitely many generators, all of even degrees. Compute the cohomology H* (F: k) of F. 7.13. In the divided power algebra F (x) over IF let (71 (x), -yp (x), . . , -y p s _ (x)) denote the ideal generated by -yp t (x) for 0 < t < s. Show that

r (x)/(71 (x), 7p(x), • • • , eYps

-1 (x))

r('-rps (x)).

7.14. The homotopy equivalence B12K(7r, n) K n) shows that we can inductively define the Eilenberg-Mac Lane spaces as K = BK n — 1). Set up the homology Eilenberg-Moore spectral sequence over a field k that has E2 -term determined by II* (K n — 1), k) and converges to H * (K n); k). Begin with K (Z,1) and see how far you can compute II * (K (Z , n); k) without needing more information to decide a differential.

8 The Eilenberg-Moore Spectral Sequence II 'By 'differential algebra' we mean algebra in the differential category; this includes homological algebra which has significance as a point of view beyond the results which constitute the field." From

[Stasheff-Halperin70]

In Chapter 7 we constructed the Eilenberg-Moore spectral sequence with which the cohomology of the fibre of a fibration can be computed from the cohomology of the base and total space. In this chapter we consider some substantial applications of this spectral sequence in situations that would be ambiguous at best with the Leray-Serre spectral sequence. The first third of the chapter concerns the problem of computing the cohomology of homogeneous spaces. If H is a closed subgroup of a Lie group G and j : H G the inclusion, then we have the fibration

and so the Eilenberg-Moore spectral sequence can be applied to compute H* (G I H;k). In many cases (Theorem 6.38), H*(BG;k) is a polynomial algebra, that is, a free, graded commutative algebra. The homological algebra TorH. (BG;k)(k,H*(BH;k)) may simleading to the computation of E2 plify sufficiently to induce the collapse of the spectral sequence. For the case k = R, this collapse is found in the work of [Cartan50] and it was conjectured to hold in general. The extent to which the conjecture holds is the topic of §8.1. In the middle third of the chapter, we give a description of the differentials in the Eilenberg-Moore spectral sequence. We begin with the suspension homomorphism (§6.2), which is known to annihilate products. [Uehara-Massey57] and [Massey581 introduced a secondary operation based on the vanishing of cup products, now called the Massey triple product. Generalizations of this secondary operation to higher orders, due to [Kraines66] and [May68], lead to a complete description of the kernel of the suspension homomorphism and the differentials in the Eilenberg-Moore spectral sequence. [Massey68/981 also

274

8. The Eilenberg-Moore Spectral Sequence II

related the triple product to higher order linking phenomena. In low dimensions Massey products were conjectured by [Stallings651 to be related to the A-invariants of [Milnor571. These relations were made precise by various authors making Massey products a useful tool in the study of low dimensional topology. A spectral sequence can be endowed with extra structure that converges to the appropriate structure on the target. For example, the Steenrod algebra structure on the mod p cohomology of the total space is carried along in the associated Leray-Serre spectral sequence (see Theorem 6.15). In the final third of this chapter, we determine a Steenrod algebra structure on the EilenbergMoore spectral sequence by giving two geometric constructions of the spectral sequence from which this extra structure follows directly. The first construction is due to [Smith, L70 1 and follows an investigation, suggested by [Atiyah621 and [Hodgkin68], of the category Top/B of continuous functions with codomain B. The Eilenberg-Moore spectral sequence can be viewed as the topological Kiinneth spectral sequence for this category. The second construction is due to [Rector70] who introduced a cosimplicial space associated to a pair of mappings > B. Cosimplicial spaces (1Bousfield-Kan721) as well E > B and f: X as the category Top/B (IlCrabb-James981) have other applications in homotopy theory. We close the chapter with a brief description of further applications of the spectral sequence and its algebraic analogues. The question of the strong convergence of the Eilenberg-Moore spectral sequence for spaces that are not simply-connected is taken up in the next chapter. —

—

8.1 Homogeneous spaces

In the applications of spectral sequences to topology, computations may involve deep geometric information (to determine the differentials; see Example 5.H) or may proceed almost miraculously via algebraic features that determine collapse at some calculable stage (see Example 5.F). We find an excellent mixture of these features when we apply the Eilenberg-Moore spectral sequence to a natural class of examples, homogeneous spaces. To H, a closed subgroup of a Lie group, G, we associate the fibration Bi GIH BH BC (as in §6.2). The cohomology of GIH is an invariant of the inclusion map—how H sits geometrically in G. A nontrivial example of the importance of the choice of inclusion is seen by considering two inclusions of U(n) in U(rt + 1). There is the usual inclusion U(n) U(n + 1) given by 1 0 0 0 ii: AI— ( . . A 0 I

275

8.1. Homogeneous spaces

that yields U(n + 1)/U(n)

s2n+1.

det

Another inclusion is given by i2, O\

0 0

A

6

J

for which we have U(n + 1) /U(n) x CP(n) ([Baum68]). We fix some assumptions that are satisfied in most of the situations of interest. Take k to be a field, H, a closed, simply-connected subgroup of G, a compact, simply-connected Lie group. Suppose further that If (G) and H(H) have torsion only of orders prime to the characteristic of k. Under these assumptions, by Proposition 6.37, H* (G; k) and H* (H; k) are exterior algebras and it follows that H* (BG; k) k[x 1 , ,x] and H* (BH; k) k[y i , , yin ], polynomial algebras on generators of even dimension greater than or equal to two. Thus Bi* is expressible as an algebra homomorphism Bi* k[x i ,.

, x rd —> k[y i ,

, ym ].

The Eilenberg-Moore spectral sequence, converging to H*(GIH;k), has as E2-term, E2 `=" TOrH . (BG ;k) (k, H* (BH; k)) that we may write as E2

Ls" TOr k[xi,•••

k [Y1, • • •

Yrn])•

From an algebraic viewpoint, this seems to be a tractable computation (see §7.3); perhaps the structure of this E2-term is tight enough to force the collapse of the spectral sequence and so determine the algebra H* (G I H; k) up to extension problems. The following collapse theorem will be considered in various forms in this section. Theorem 8.1 (the collapse theorem). If k is afield, either of characteristic zero or of characteristic p and H* (G) has no p-torsion, then the Eilenberg-Moore spectral sequence for G I H BH BG collapses at the E2-term and so, Bi as a graded vector space,

H* (G 1 H; k)

Tor Tr- (Bc ; k)(k, H* (BH; k)).

Before we describe the various approaches toward proving such a theorem, we consider the motivating case due to [Cartan50]. Theorem 8.2. If H is a closed, simply-connected subgroup of G, a compact, simply-connected Lie group, then H*(G1H; lit )

TOrw.(BG ; T)

H* (BH; 1)).

276

8. The Eilenberg -Moore Spectral Sequence II

SKETCH OF PROOF: We exploit the analytic structure of Lie groups and compute their real cohomology using de Rham cohomology (see the books of [Warner831 or [Bott-Tu82] for details). The de Rham theorem tells us that HLR (M) R H* (M; R) as algebras over for a manifold M and so no structure is lost in this choice. Let l* (M) denote the de Rham algebra of differential forms on the manifold M. Following essentially the same proof as given for Theorem 7.14, it can be shown that HLR(G/H) R Toro* ( B0)(R, St* (B H))

The de Rham algebra makes sense for BC and BH because these spaces can be approximated to any dimension by homogeneous spaces ([Steenrod511). To prove the theorem, it suffices to show that Toro* (BC) (R, (B H))

TorHL R (BC )

HLR (B H))

and so the Eilenberg-Moore spectral sequence collapses. Since the field R has characteristic zero, H* (G; R) is an exterior algebra and H* (BC ; R) is a polynomial algebra. Write kileR(BG) `L' 11k[ui • • • Consider the mapping h: H-44'eR (BG) SI* (BC) defined by choosing a representative for each ui (for example, the harmonic representative of Hodge theory). Since St* (BC) is a graded-commutative, differential graded algebra, we can extend the mapping to an algebra mapping HIR(BG) > 1l (BC) that induces the identity on homology. We can do the same for BH . Though the diagram Bi*

HLR(BH) h

Bi*

S2* (BH)

need not commute without further assumptions, the algebra mappings endow St* (BH) with two HLR (BG)-module structures that coincide on homology. By applying Corollary 7.7 a number of times, we establish the desired isomorphism and the theorem. (See the paper of [Baum-Smith671 for further details and a generalization.) Theorem 8.1 refers to any field of coefficients and so it is a generalization of Cartan's theorem. We present three versions of Theorem 8.1. The first version is due to [Baum68] and depends on the explicit algebraic structure of Tork[„,,... (k, k[yi, , ym ]). The geometric input is a theorem of [Bore153] on the role of the maximal tonis of the subgroup H in the computation of Tor H* (BG;k)(k, H* (BH; k)). Though Baum's theorem requires a supplementary algebraic condition for collapse, it remains useful and we apply it to

8.1. Homogeneous spaces

277

compute the mod p cohomology of the real Stiefel manifolds, H* (Vk (Rn); for p, an odd prime. The range of applicability of Baum's method remains an open question. In the second version, we present work of [Gugenheim-May741 who exploited the simplification afforded by a theorem of [Baum681 on the maximal torus, and constructed special resolutions as well as explicit mappings for tori, C* (BTn; k) H* (Brn;k), that induce the identity on homology. This implies a collapse theorem in the manner of the proof of Theorem 8.2. The resolutions considered in their work are based on a definition of Tor and Ext for differential algebras that relates more closely to the homology of the modules and algebras involved. These new definitions allow the construction of more computable resolutions. The third version of Theorem 8.1 is based on the freeness of H* (BC; k) and examines mappings, H* (BC; k) C* (BC; k), which can be constructed because we have a free algebra basis, but are obstructed as algebra mappings by the lack of commutativity of the cup product on C*(BG; k). The system of higher homotopies needed to remove the obstructions is catalogued by the bar resolution and its coalgebra structure. Motivated by a similar phenomena in the study of H-spaces, [Stasheff-Halperin70] began a study of coalgebra mappings of the bar construction as a system of higher homotopies. This line of ideas culminated in the powerful collapse theorem of [Munkholm74]. (The statement of Theorem 8.1 is due to [Wolf77] and follows by similar means.) With the development of the topics in §8.1, we settle a special case of the collapse of the Eilenberg-Moore spectral sequence. From the examples of homogeneous spaces and the computation of the cohomology of certain two-stage Postnikov systems, one might conjecture the collapse of the Eilenberg-Moore spectral sequence where E2 TorA (B, C) and A, B and C are free graded commutative algebras. [Hirsch] posed this problem for two-stage Postnikov systems and [Schochet71] gave a counterexample to Hirsch's conjecture. The Eilenberg-Moore spectral sequence is a natural tool for the study of two-stage Postnikov systems (see the papers of [Smith, L67, 71]). Baum's thesis

The homotopy-theoretic computation of H*(GIH;k) was begun in the papers of [Samelson411, [Koszu1501, [Leray501, and [Cartan50]. These developments were focused on the real cohomology of GIH. Among the structural features revealed in these papers, a key role was played by maximal tori and maximal rank subgroups. Recall that the rank of a Lie group, G, may be defined as the number of algebra generators of H* (G; Q). When G has no torsion of order a power of the characteristic of the field k, this rank is also the number of algebra generators of H- (G; k) or H- (BG;k). [Bore153] computed the cohomology of G I H for more general coefficients under the following conditions:

278

8. The Eilenberg -Moore Spectral Sequence II

Theorem 8.3. Suppose H is a closed subgroup of G and rank G = rank H and that H* (G) and H* (H) have no p-torsion where p = char k. Then (1) As an H* (BC; k)-module, induced by the algebra homomorphism Bi*, H* (BH; k) is finitely-generated and isomorphic to the extended module

H* (BC ; k) 0 H* (G I H; k). Bi*

H* (BG; k) H*(BH; k) H* (G H; k) k is coexact, that is, j* is onto and ker j* is generated as an ideal by Bi* (H± (BG; k)).

(2) k

BH PROOF: The Leray-Serre spectral sequence for the fibration G/H BG implies that H* (BH; k) is a subquotient of H* (BC; k) 0 H* (C /H; k). Since k[xi, ,x] H* (BC ; k) is Noetherian, it follows that H* (B H; k) is finitely-generated over H* (BG;k). Since rank G = rank H, H'(BH;k) is a polynomial algebra on the same number of generators as H* (BC ; k) and is also finitely-generated. It follows (Variski-Samue158/601) that H* (BH;k) is free as an H* (BC ; k)module (Corollary 3.10 of [Baum68]). Apply the Eilenberg-Moore spectral sequence to the fibration Bi. Since H* (BH; k) is free over H*(BG; k), (BG;k)(k, H* (BH; k)) k

(BG;k)

H* (BH; k)

and the spectral sequence collapses to the column E'* . Thus H* (BH; k) H* (BC ; k) 6_4 k

H* (Bc ; k)

H* (BH; k)

H* (BG; k) Ok H* (G I H; k).

The coexactness of the sequence follows from the fact that im j* = and k 0 Fr* (BG;k) H* (.13 H; k) is the set of indecomposables with respect to the H* (BC ; 0-module action, as required. The following aspect of Lie group structure is central to the computations of H* (G I H; k). Definition 8.4. Let G be a compact connected Lie group. A subgroup T is called a maximal torus of G if T is abelian, compact and connected (from which it follows that T is isomorphic to S i- x x (n times) for some n) and if U is another such subgroup with Tc Uc G, then T = U. From the classical theory of Lie groups (see [13ore1551 or [Mimura-Toda911) we know (1) All maximal tori are conjugate to one another (2) Given a torus subgroup T' of G, there is a maximal torus T with T' c T C G.

8.1.

Homogeneous spaces

279

(3) rank T = rank G. (4) If NT is the normalizer of T in G, then T has finite index in NT. The finite group, W(G) = NT IT, is called the Weyl group of G. (5) The Weyl group acts on H* (BT) = qv). ,un j, deg vi = 2 for all j. If T G is the inclusion and H* (C) has no p-torsion for p = char k, then Bi* : H* (BC; k) H* (BT; k) maps H* (BC; k) isomorphically onto k[vi, ,v n ] w(G) , where k[vi, ,un ] w(G) denotes the ring of invariants of the Weyl group action on H* (BT;k). [Baum68] reduced the problem of the collapse of the Eilenberg-Moore spectral sequence for computing H*(GIH;k) to the case of H*(GITH;k) where TH is a maximal torus in H.

Lemma 8.5. Let

TH C H

be a maximal torus of H. Then, Pr all i < O,

To r

(13G;k)(k7 H* (BH; k)) {0} if and only if TorH , (BG;k)(k, H* (BTH;k)) {0}. PROOF: Since rank TH = rank H, by Theorem 8.3, H*(BTH; k) is isomorphic to H*(BH;k) 0 H*(H ITH; k) as an H* (B H; k) -module. The H* (BC ; k)-module structure on H* (BTH ; k) is derived from the inclusion TH CHCG and so it factors through H* (B H; k). Thus, as an H* (BC ; k)module, H* (BTH;k) is still isomorphic to H* (B H; k) O H* (H/TH; k) and so, as vector spaces, (k 7 H* (BTH; k)) =TorV, (BG;k) \(k,H* H; k))0H* (H/TH ; k). Tori'* H* (BG;k)

Since H/TH is connected, the result follows. Corollary 8.6. Let TB C H be a maximal torus of H. Then the Eilenberg-Moore spectral sequence, converging to H* (C /H; k) for the fibration GIH BH BC, collapses at the E2-term if and only if the corresponding spectral sequence for G ITH BTH —> BC, converging to H*(GITH;k) collapses at the E2-term.

PROOF: It suffices to observe that the morphism of spectral sequences induced by TH C H respects the isomorphism of the proof of Lemma 8.5 as well as the splitting as a tensor product. Thus E2 = Eco for G/TH Bi Bi BTH BC implies E2 = Eco for G/H EH BC by natural ity. The converse follows from the injection of TorH. (BG;k)( 1C1 H* (B 11 ; k)) into TorH.(BG ; k ) (k, H* (BTH;k)) as a factor with the quotient H* (H/TH; k) concentrated in E2° '*. With Corollary 8.6 we can prove the following technical lemma that leads to Baum's collapse theorem.

280

8. The Eilenberg-Moore Spectral Sequence II

Lemma 8.7. Tor ik** (BG;k)(k, H* (B H; k)) = {0} if i < rank H — rank G.

PROOF: It suffices to prove this for TorV„(BG•k)(k, H* (BTH; k)), Ti C a maximal torus. Let T C G be a maximal torus of G containing TH and A = T/TH with 7: T —> T ITH, the canonical projection. The maps A T —> G induce the maps BA <— BT —> BC, which on cohomology provide H* (BT; k) a module structure over H*(B A; k) as well as over H* (BG; k). As an algebra, H* (BT; k), is isomorphic to H* (B A; k) 0 H* (BTH ; k) (just count the S 1 factors) and so H* (BT; k) is a free H* (B k)module. Since rank G = rank T, Theorem 8.3 implies that H* (BT; k) is also a free H* (BC ; k)-module. In the presence of free modules over pairs of related algebras, we have a change-of-rings theorem for Tor. We leave the proof of the following elementary fact to the reader. H,

Fact: Suppose A, B, and C are algebras over a field k. If M is a right A-, right B-module, N is a left A-, right C-module, and L a left A-, left B-module, and Tor71 (M, N) = {0} = (N, L) for n> 0, then Tor*Bec (M 0 A N, L) Tor'A e, B (M, N 0c L).

In our case, A = k, B = H* (B A; k), C = H* (BC ; k), M = N = k and L = (BT; k). Because every algebra in sight is graded commutative, we can apply the change of rings to obtain the isomorphisms, TOrH*(BA;10

(k, k ®H*(BG ; k) H* (BT; k))

Torn- (BA ;k) ,DTP, (BG ; k) (k, H* (BT; k)) Tor H*(BG ;k)(k,k OH* (BA;k)

(BT; k)).

Since k0B - - (s A;k)H* (BT; k) is isomorphic to H*(BTH; k) as an H* (BC; k)module, we finish the proof of the lemma by observing that To r

( ; k)(k, k ( 5-D H- (sc ; k) H* (BT; k)) = {0}

when j < rank H — rank G. This follows because H* (B; k) is a polynomial algebra on rank G — rank H generators. In homological degrees i < rank H — rank G, the Koszul resolutions for modules over H* (BA; k) are trivial (Corollary 7.23). We focus on some homological algebra: What can be said about the structure of Tor k [„,,... , ym ]) in the ease where k[yi , , ym ] is a . . . , oc ri] -module through an algebra homomorphism Bi* : k[xi, . . . , oc n ] —> k[Y1 , • • • 'MA?

281

8.1. Homogeneous spaces

Definition 8.8. Suppose A is a connected, commutative, finitely-generated algebra over k that is concentrated in even degrees. A presentation of A is a coexact sequence k[wi,

,w 8 ] —>

,vt]

f

E

for which the induced mapping g = 10 g g: k

k[wi, • • • , ws] + —> k

ker f

is an isomorphism of graded vector spaces over k.

The sequence being coexact means that ker E is generated as an ideal by , g(w,)}. In general, the sequence of maps due to the fibration {f (v i ), , f (vt)} and ker f is generated by {g(tvi),

Be

H* (BG; k)—>H* (BH; k) —>H* (G I H; k),

does not comprise a presentation of H* (G I H; k) nor is it coexact. However, as proved in the next few results, the conditions described in the definition provide some control of the algebra of the E2-term of the associated spectral sequence. Given a presentation of an algebra ,

k[vi ,... ,vt ] L

the elements {g (w 1) , , g 0_1)8 11 in k [y1, . . . , vt] minimally generate the ideal ker f in the following sense; no proper subset of the g(lvi ) generates the ideal ker f. This follows by the vector space isomorphism :g = 1 ® g g : k{wi,

—> kgk[ w i ,...

, tvs ]

ker f

ker f/ (krwi,

,w, p+ ) . ker

When we have such an isomorphism for an ideal I in k[ni , ,7.1t ], we say that the set {g 1) , g(w8) } forms a nonredundant set of generators for I. In the next proposition we connect presentations of an algebra and the homological algebra of polynomial rings. Recall that the graded vector space of indecomposables of a connected algebra B is given by Q(B) = B+ I B+ • B+ ko B B+, where k is a B-module by the augmentation E : B k and B+ = ker E. The vector space of homogeneous elements of degree i in Q(B) is denoted by Qi (B).

282

8. The Eilenberg-Moore Spectral Sequence II

Proposition 8.9. If A is a connected, commutative, finitely-generated algebra over k that is concentrated in even degrees, and

and

k[wi,

, ws ]

k[vi,

, vt ] —> fl A —> k

k[x i ,

,x] —> k[y i ,

f2 , y q ] —> A —> k

g2

are two presentations of A, then Tork[wi,••.

(k, k[vi,... , vt])`"="

(k,k[yi,.

, yq ])

as bigraded algebras over k. Furthermore, for any i, dimk Q i (k[w • • • ,w 8 ]) =

(k[vi, • • • ,vt])

dimk

dimk Qi (k[x i ,...

,x]) — dimk Qi (k[Yi, • • • ,Yq])•

PROOF: We first treat a special case. Suppose that yi and 9 2 have the same domain and range, that is, fi = f2 = f, but gl g2. at

,W]

22

>

'k[vi, • • • , vt]

A

k.

Following §7.3, we compute Tor k [wi ,... ,„ s i (k, k[v i , ,v]) in each case via , es ) the Koszul resolution. Let Lc , * = A(ei, k[vi,. ,vt] and LC , * = A(El, ,e ) e)k[v], • • • ,vd denote the Koszul complexes for the given data. The differentials on each complex are determined by the mappings gi and 92 that induce the k[w i ,. , w8 ]-module structures: d(1 0 t(1 ) = 0 = d(1 e y3 ), d(ei vi) = 1 gi(wi) • vi

and

d(Ei vi) =10 g2(wi) • v 7 .

Since the two pairs of mappings (gi, f) are presentations of the algebra A, the sets {gi (wi), • • • (w8)} and {92(W1), • • • ,g2(w s )} are nonredundant ker sets of generators of k{w i , • • • ,u)s } f. Let k Ok[wi,••• ,w s] ker f denote the reduction of gi modulo the module actions. Then the sets -Di (wi) } , for i = 1,2 and 1 < j < s, are vector space bases of k ker f. It follows that we can write g 1 (w1 ) = E1g2 (u 1 )u 11 for up c ker f. Using the matrix of elements u11 E ker f, define a mapping of Koszul complexes given by : L" —> L" , 0(1 u) = 1

u, 0(e3 0 V) =

uji V,

where we extend as a mapping of modules over k[vi,... , vt ]. The mapping 0 commutes with the differentials: Od(ei g V)

—

0(1 0 gi (w1 )17)

=1

gi (wi )1/' =

1

11

0 (g2(wi)u 11)V

10 g2(w/)(ui/V) d

= dO(ei ® V).

/

uji V)

283

8.1. Homogeneous spaces

In k

,,] ker f, {(wi)} are bases. The change of

basis matrix

(w)e(u1i and so (e(wii) is an invertible matrix.

is given by :gi(w1) =

It follows that Q9(e1 ) = Eie(u]ii) and so QO is onto. Since L* , * is a free graded commutative algebra, Qû onto implies û is onto. As finite dimensional vector spaces over k, LP , q and LP , q have the same dimensions and so O is an isomorphism, and 0 induces an isomorphism on homology. For the general case of two different presentations, build an intermediate complex. First take the composite k[vi,... ,v as

h

k[yi, • Yq]

flef2

> A e A —> A

and then choose g3 for r = s p k[u t ,

, ur]

k[zi

, vt]

f3 k[Y1, • • • Yq]

A —>

so that the Koszul complex for this presentation has an acyclic factor and a factor with homology Tork[wi ,... ,v i ]). By a symmetric construction, one can choose a g4 with the same domain and range and having an acyclic factor and homology given by Tork[„,,... ,„,](k,k[gi... , yq ] ). By the first case we have established the theorem. To prove the assertion about the dimensions of the Qi, one argues by comparing dimensions of indecomposables for the two Koszul complexes used in the argument for the general case. With this proposition, two homological invariants of the algebra A emerge. First, we have the bigraded algebra J"(A) =

(k,k[vi,.. • ,vt])

constructed from any presentation of A. Though J(A) is not functorial in A, it is determined up to isomorphism and furthermore, J(A)°' * A. Secondly, define the integers dfi (A) = dimk Qi (k [wi , • • • , ws ]) — dimk Qi (k[vi, • • • ,vt])

and the deficiency of A as df (A) = dfi (A). Riaum681 identified the class of algebras for which J(A)i , * = {0} for i < 0 or, equivalently for A finite-dimensional, with df (A) = O. He refers to such algebras as E-algebras and they can be characterized by being generated, as an algebra, by a regular sequence, that is, a set of elements al , , at in A+ so that, for i > 1, the class [ai ] in A/(a , , ai_i) is not a zero divisor. Polynomial algebras are the simplest examples of E-algebras. The class of E-algebras has been studied in algebraic geometry and research on local rings where they are called complete intersections. In topology, E-algebras sometimes occur as the cohomology of a space—see the work of [Smith, L82 1 and [Vigué-Poirrier95]. The new invariants of A, J*, * (A) and df (A), combine with coexactness to give us a toehold on the structure of Tork[,,,,... „„,,1(k,k[vi,

284

8. The Eilenberg-Moore Spectral Sequence II

Theorem 8.10. If the sequence of algebra mappings, g

k[wi, • • • w ] — > k[tli, • • • , vt] —>

A

—

> k,

is coexact (but not necessarily a presentation), then Tor k [wi ,..

k[vi , • • • • vt])

Eo J (A),

as bigraded algebras, where E = A(e i ,

, er ), is an exterior algebra on the elements e i of bidegree (-1, s i ). Furthermore, for each i,

dim k E+1,i = dimk (k[w i ,

,w 8 ]) — dirnk Qi (k[vi ,

, vt ]) — dfi (A).

that ai = g(wi) and the set {a} generate ker f , not necessarily in a nonredundant manner. Suppose that a, = a,_iu,_i ± • • • + Let (L 8,8 , d) be the Koszul complex A(ei, .. • , es) 0 , vt] with differential given by d(1 u) = 0, d(ei = 1 0 au. Let (L 8 ' 8 , d) be the bigraded algebra with L = A( 1 , , Es ) k[v i ,... , vt ] and differential given by d(ei Ou) = 1 0 au for 1 < i < s, d(es 0 1) = O = d(1 u). Define a homomorphism : L 8,8 —> L* , * by 0(1 O u) = 1 0 u, V)(e i ®1) = ei 01 for i < s and, finally, 0(e 8 0 1) = es 0 1 + Es-1 0 us _i ± • • • ± ë O u i . Then 0 commutes with the differentials and it induces an isomorphism on Tor. The theorem follows. PROOF: Suppose

Notice that the bigraded algebra E8,8 in the theorem measures the failure of the coexact sequence to be a presentation. Thus, if the coexact sequence contains superfluous algebra generators (g is not an isomorphism), then E has a generator for each wi that is superfluous. In the case of a homogeneous space and the absence of torsion at the characteristic of k, Theorem 8.3 implies that we have the sequence of algebra mappings: Bi*

j*

H* (B H; k) —> H 8 (BC; k) —> H* (G1 H; k) —> k.

Here H* (BH; k) k[w i , ,w s ] and H8 (BC; k) k[v i , , vt ]. The sequence need not be coexact, however. In order to define deficiency in this case, we focus on Bi*. For any mapping f : A —> .13 of commutative algebras, we define BlIf to be the quotient BI f(A+).B. With this notation, the sequence Bi

H* (BG;k) —> H* (BH; k) H* (BH;k)11Bi* —> k

is coexact and so Theorem 8.10 applies. We define the deficiency of H in G by df l (H, G; k) = df l (H*(BH;k)11Bi*) and df (H, G; k) =

df I (H, G; k).

285

8.1. Homogeneous spaces

Theorem 8.11. For H, a closed, simply-connected subgroup of a compact, simply-connected Lie group, G, if df (H, G; k) < 2, then the associated Eilenberg-Moore spectral sequence collapses, that is, as bigraded algebras, E ,* (H*(G1H;k)) -= Tor;i! (BG;k) (k, H* (B H; k)).

PROOF: Theorem 8.10 implies that Torx.(BG ;k)(k,H * (BH; k)) is isomorphic as a bigraded algebra to E (H* (BH; Be) where E is an exterior algebra generated by E -1, * . Furthermore, dimk E -1, * = rank G - rank H - df (H, G; k) > rank G - rank H - 2.

If J1, *(H* (BH;k)11Bi*) {0} for 1 < -2, then E J(H* (BH; k)I1Bi*) will be nontrivial in homological degrees less than rank H - rank G by tensoring with the appropriate numbers of exterior generators. However, this contradicts Lemma 8.7. Therefore, as an algebra, TorH. (BG;10(k, H* (BH; k)) is generated by elements of bidegree (p, q) with -2 < p < 0. Since the EilenbergMoore spectral sequence is a spectral sequence of algebras, the differentials are determined by their values on the generators.

Notice that d2, of bidegree (2, -1), is zero—the generators all lie in even total degrees, which follows from the form of the Koszul complex. The higher differentials, of bidegree (r, 1 - r), take the generators in E -2, *, E -1, * and E° , * to zero. Therefore, E2 = E00 and the theorem follows. As a sample application of Theorem 8.11, we compute H* (VI (It') ; for p, an odd prime. Recall (Example 5.H) that V/ (Rn ) denotes the Stiefel manifold of /-frames in Rn . We can represent Vn _1 (Rn ) as the homogeneous space S 0 (n)/S0(/) where SO(/) is a subgroup of SO(n) by the mapping i : A 1-)

/n0

0 A

(Li _l is the (n - 1) x (n - 1) identity matrix). We take n and / to be even and consider Be H* (BSO(n); H* (BS0(1); If subscripts indicate )

.

286

8. The Eilenberg -Moore Spectral Sequence II

dimension, then Bi* is determined by Bi* : 7p[X4,X8, • • • ,

X2n-4, Fp [Y4, Y8, • • • , Y21-4, Bi* ( x4 ) = y4i , for 1< i < (112) — 1,

Bi (X21) = (0 2 Bi* ( x4 ) = 0, for (1/2) ± 1 G

(n/2) — 1,

Bi* (x I) = O.

It follows directly that H* (BS0(1);7 p )11Bi* = I [yi]l(yii) 2 and the presentation Tp [W21]

Ep [v]

p

[y I(V) 2

shows that df (SO (/), SO (n); 7p) = O. By Theorems 8.3 and 8.11, we obtain H* (Vn _ 1 (Rn);Fp ) E" J(H* (BSO(1);4)11 Bi*) A(e21+3, • • • e2n - 5, en — i') 0

The factor A (e2i+3, • • • , e2n-5, en _i') comes from the ke rnel of Bi* . The other factor J(H* ( BS° (1 ); 7P)// 13 i* ) is TorF, [x 4 ,... ,s2/] (FP , Fp [Y4, ... , Yi i ])• The Koszul complex for this Tor splits into an acyclic factor and the complex d) with d(e21_i 0 1) = 1 (0 2 . The homology of this (A(e21-1) 0 7p small complex contributes Fp [V]/(y/') 2 . Another approach to differential homological algebra 0

In the previous section, the Eilenberg-Moore spectral sequence was used to compute H* (G I H; k) in many cases. Baum's algebraic condition, df (H, G; k) < 2,

insures that plenty of 'holes' are strewn about the E2-term of the spectral sequence and so it collapses. The limitations of this approach for proving a more extensive collapse theorem are clear—the condition df(H, G; k) < 2 is not generic. In this section and the next we consider more general situations in which the spectral sequence collapses. If H is a closed subgroup of a compact Lie group G, as in Theorem 8.1, then we seek conditions that guarantee TOr H* (BG;k)(k,

H* (BH; k))

Tor c* (BG;k)(k , C* (B H; k)).

Corollary 7.7 and its application in the proof of Theorem 8.2 provide an approach. When there exist algebra mappings C* (BC

H* (BG; k)

or H* (BG; k) —> C* (BG; k)

and C* (B H; k) —> H* (EH; k) and H* (B H ; k) —> C* (BH; k)

8.1. Homogeneous spaces

287

that induce the identity on homology and commute with respect to Bi*, we obtain the desired collapse. An obstruction to the existence of such mappings is the fact that H* (X; k) is a graded commutative algebra, but C* (X ;k) is commutative only up to chain homotopy. The success of the proof of Theorem 8.2 relies on the graded commutativity of the de Rham complex of a manifold that allows us to construct honest algebra mappings. Our next approach toward a proof of Theorem 8.1 is in three parts; we first redefine the functor Tor, following [Gugenheim-May74] and [Felix-HalperinThomas95], in order to exploit resolutions that make clearer the relationship between Torr (M, N) and Tor H (r )(H(M),H(N)). Next we consider the failure of commutativity of the singular cochain complex of a space and how it is measured by the cup i -product. This cupi -product, together with the free commutative algebra H* (BC; k), leads to a resolution of the appropriate form. Finally we apply this special resolution to the geometry of homogeneous spaces and obtain a collapse theorem. Definition 8.12. Let (r, dr) be a differential graded algebra over afield k. A -module (X, d) is semifree if X is an increasing union of submodules X(1) c X(0) c X(-1)c c X(—n) c c X such that X (1) and X (—n)I X (—n +1) are free over r on a basis of cycles. A semifree resolution of ar -module (M, d) is a semifree module (X, d) together with a homology isomorphism of r-modules a: (X, d) —> (M, d).

The key features of semifree resolutions follow from the filtered structure that is part of the definition. We filter (X , d) by F— PX = X(—p). The associated graded module E* (X, F) = X (—p)1X(—p +1) X — P, * Ør for p> 0 and for some graded k-vector space X — P, *. The quotient mapping X (—p) —> X — P, * F splits to give X (—p) X (—p + 1) ED X -P, * F. The differential on E0*,* takes the form 1 0 dr and hence the restriction of d to the basis satisfies dl : X — P, * —> X (—p + 1). In the differential homological algebra of Chapter 7, the total differential on a resolution has the form dtotai = do + dl, that is, the total differential is the sum of an internal differential and a resolution differential. In the case of a semifree resolution, the differential may be further refined; d = c—dr>0 iir where dr (X (—p)) C X(—p + r). We associate the usual spectral sequences to the filtered module (X, F). By the definitions, E* (X) = H* ( ,1) and E* (X) L' V.* H(r) for i < O. If the sequence E l (d • • • —>

Ef'* (X) —> El* (X) —>

> E* (X) —>

(M) —> 0

is exact, we say that X is a resolution of M. Furthermore, if each Er (X) is a flat HP-module, then we define, for (N,dN), a left differential graded

288

8. The Eilenberg-Moore Spectral Sequence II

module over r, Torr (M,

H(X 0 r N).

The induced filtration on X or /V, given by FP (X or N) = FX or N, yields a spectral sequence converging to Torr (M, N). By the Kfinneth theorem, E2 TOrH(r)(H(M), II(N)),

where Tor is the classical functor for graded modules over Hp. If we take Y. —> M 0 is a proper projective resolution of M over (r, dr )• we can compare it with a semifree resolution by using the projective property and Corollary 7.7 to obtain an isomorphism between Torr (M, N) and Torr (M, N). We next state a theorem that describes the computational advantage of this new functor Tor. The proof is by a direct construction given in [GugenheimMay74, p. 121. Theorem 8.13. Suppose M is a right differential graded module over r and H(M) has a projective resolution over H (F) of the form • • • —>

0H(r) —> )7+1-* ex(r) —> • • • —>

7"0B-(r)

H(M) —> O.

Then there is a differential on the filtered F-module, X = X" 0 r and a homology isomorphism a: X —> M so that X is a resolution of M whose associated El -term agrees with the given resolution.

To apply this framework to homogeneous spaces, we study the singular cochain algebra. The cup product on H* (X; k) is graded commutative, while C* (X; k) is only commutative up to homotopy. The failure of graded coinmutativity on C* (X; k) gives rise to cohomology operations as in the classical construction of the Steenrod operations. We realize the chain homotopy for the cup product by the cup i -product (§5.3) and generalize this structure to differential graded algebras as follows: Definition 8.14. A differential graded algebra (r,,u, d) over k is said to have a cup' -product if there is a k-linear mapping for all p and q —1: FP Øk rq

FP+ q-1

satisfying the Hirsch formulas; if we write jt(a,b) = a • b, then d(a

ai

• a — d(a) b — (-1) 1 '1 a —1 d(b), + (-1)1b1 lel . b. (a • b) —1 c = (-1) 1 '1 a • (b h) = a • b — (-1)

289

8.1. Homogeneous spaces

For singular cochains with coefficients in k = F2 the cuppproduct determines an operation on the cocycles of X; for each s, z 2s—i (x ;72),

Sq l : Zs (X:F2)

defined by Sq l (z) = z —1 z. This passes to a homomorphism Sq l : Hs (X:72)

H2s—i (X; 72 )

and Sq l = Sq8-1 , the operation in the mod 2 Steenrod algebra (see the construction in [Steenrod-Epstein621). The first of the Hirsch formulas provides a chain homotopy between the mappings 12 and 12 oT:r0F —> F. The second formula can be understood to say that for each c E r, the mapping a a c is a derivation with respect to tt. Note that the homology of a differential graded algebra with cupi -product is graded commutative. Furthermore, if u: F —> H(r) is an algebra homomorphism that induces the identity on homology and commutes with cupi -products, then v must annihilate cupi -products. We will soon see that the condition that a differential graded algebra mapping r H(r) annihilate cupi -products is sufficient to induce the isomorphism we are seeking for the collapse theorem. Suppose F is a differential graded algebra with cup i -product such that H (F) is a polynomial algebra on even degree generators, H(r) k[x1, . , x rd. Let M = k; by Corollary 7.23, the Koszul resolution, CH (11) = At{ai }) H(r), provides a resolution of k of the form given in Theorem 8.13. If we let K(F) = r with the obvious augmentation, E: k(r) k, then Theorem 8.13 implies that there is a differential on 1C(F) such that K.(F), is a resolution of k and E1 ((F)) is the Koszul resolution of k over H(r). [Gugenheim-May74] give the differential on 1C(F) explicitly. The vector space generators for A(ct i , , an ) are indexed over the set of sequences I = <2 < • • • < ii ). We write ai = ai i cti, • • • aip . If H(F) k[xi, • • • xn] and, for i = 1, , n, bi G F is a representative for xi E H(r), then we associate to each sequence I the element ,

= (• • • ((bi, —1 bi,) —1 bi 3 ) (b (i1) =

bi ,

i • • •)

i bip)

and bo = 0). The differential on r(r) takes the form ±(I,

d(ai) = — Jcl

where J is a subsequence of I and I — J is the complementary subsequence. The sign is given by ±(I, J) = (-1)P— r+ (J) where r is the length of J and E(J) = Etr_,(J(t) — t). With this definition, it is shown that dd = O. With an explicit semifree resolution, [Gugenheim-May741 prove the following algebraic collapse theorem.

290

8. The Eilenberg -Moore Spectral Sequence II

Theorem 8.15. Suppose A and F are differential graded algebras over k that have cup i -products. Suppose f: F > A is a differential graded algebra mapping that commutes with the cupi-products. If r is augmented, H(r) is a polynomial algebra, and there is a mapping g: A —> H (A) of differential graded algebras with H(g) — id such that g annihilates cupi -products, then, as graded vector spaces, Torr (k, A) TorHm(k,H(A)). —

The proof reduces to examining the induced differential on K(r) (gr A where the expression for the differential on 1C(r) and the fact that g annihilates cupi -products imply the isomorphism. Having developed the algebra to this point, we add a geometric fact that turns the key for the desired collapse theorem for homogeneous spaces. The proof of the next proposition is found in the appendix of [Gugenheim-May741 and it follows by applying the simplicial techniques in [Eilenberg-Mac Lane531. Proposition 8.16. Let I' = x•••x (n times). For any commutative ring R with unit, there is a morphism of differential graded algebras over R C* (BT ; R) —> H* (BT 71 ; R) that induces the identity on homology and annihilates cupi-products. Theorem 8.17. If H is a closed, simply-connected subgroup of a compact, simply-connected Lie group G, satisfying the conditions given in Theorem 8.1, then, as a graded vector space, (G 1 H; k)

Tor II- (BG ; k)(k, H* (BH; k)).

PROOF: Let TH C H be a maximal torus for H. By Corollary 8.6, it suffices to prove that H* (G ITH; k) is isomorphic to TorH. (BG;k)(k , H* (BTH; k)). But TH T'2 for some n and so, by Proposition 8.16, there is a mapping C* (BTH; k) —> H* (BTH; k) that preserves products and annihilates cup, products. By the assumptions on G and H in Theorem 8.1, the rest of the assumptions of Theorem 8.15 are satisfied and so Torip((sa ; k)(k, H* (BTH; k))

and the theorem is proved.

Tor c (sc ; k)(k,

(BTH; k))

111

[Gugenheim-May74] developed their results over a ring and so they apply more generally with the correct flatness assumptions. The dual theory for the functor Ext is also developed. Finally, the assumption of a finite polynomial

8.1. Homogeneous spaces

291

ring for H(F) plays no role in the proof and so it is possible to study the Eilenberg-Moore spectral sequence for a two-stage Postnikov system:

(1Ki

S1Ki

1

1

E

PI K2

K1.

Here K1 and K2 are taken to be Eilenberg-Mac Lane spaces and 7C' the path-loop fibration. When k = 72 the cohomology of K, is polynomial (Theorem 6.19) and the homological algebra of [Gugenheim-May741 may be applied. [Schochet71] studied an important example that turns on the cupi -product structure. Let z and .) be the fundamental classes in H2 (K(7/27117/27, 2);1F2) and take K, = i -- 3: KR72E /7,/2Z, 2) —> K(z/n, 4).

e

[Schochet711 showed that there is a class in E2-2 ' 17 of the associated EilenbergMoore spectral sequence on which d2 is nonzero. This gave a counterexample to a conjecture of [Hirsch] that asserted an additive isomorphism H* (E;T 2) 11* (K i ; 72) and H* (K2 ;IP9 2 ) are polyTorH. ;F2) ( 7 2, H* (K2; k)) because nomial algebras. Suppose X is a space with exactly two nonvanishing homotopy groups:

{ G1, 7, (X) =

G2,

{0 } ,

if r = n, if r = n + k, otherwise.

Up to homotopy, X can be realized by the pullback over a mapping of Eilenberg-Mac Lane spaces of a path-loop fibration: K(G2, n

E

K(Gi,

k) = K(G2,n + k) PK (G2 , n + k +1)

K(G2, n

k + 1).

The two-stage Postnikov system, X, depends on the choice of [K]; [k] E [K(Gi,n),K(G2,n + k +1)] = Hn±k±l(K(Gi,n); G2).

292

8. The Eilenberg -Moore Spectral Sequence II

Thus [k] can be interpreted as a primary cohomology operation. In §9.1, we define higher order cohomology operations and find that the space X is the universal example for the secondary operation based on [n]. This motivates the study of H* (X; R). [Kristensen631 studied such spaces using the LeraySerre spectral sequence and the action of the Steenrod algebra (Theorem 6.15). [Smith, L67'] and [Kraines73] applied the Eilenberg-Moore spectral sequence in their study of two-stage and higher stage Postnikov systems. Extending the functor Tor

g

We return again to the main idea of Corollary 7.7—when f : A —> F is a morphism of differential graded algebras, g: M M' and h: N —> N' are morphisins of modules, and all of these mappings induce isomorphisms on homology, then TorA (114-, N) Torr (AV, N') Thus Tor only 'sees' the chain homotopy type of the algebra and modules. In this section we extend Tor to accept morphisms that are algebra and module mappings up to chain homotopy. The idea that properties up to homotopy are the more fundamental notion appears crucially in the study of H-spaces. For example, a moiphism of chain complexes need not be a morphism of algebras or modules but induces such a moiphism on homology. Perhaps the simplest example is the homotopy associativity of the loop multiplication on (2 (X, x 0 ). Here the multiplication is associative up to homotopy, has inverses up to homotopy, and a unit up to homotopy. More dramatically, we know that there is a multiplication on S 7 (the unit Cayley numbers), which is not associative. We can ask in reverse whether it can be deformed into an associative multiplication. [Sugawara571 and [Stasheff63] constructed a homotopy invariant way to study associativity. The notion of An -structures ([Stasheff63]) stratifies possible 'degrees' of associativity—an A3-space is a homotopy associative H-space and an Ace -space is homotopy equivalent to an associative H-space. [Clark65] extended these notions to study the commutativity of the loop multiplication on WC when X is an H-space. The analogous structures for differential graded algebras and modules over them were identified by [Stasheff-Halperin70] who proposed the following approach to the problem of the collapse of the EilenbergMoore spectral sequence for homogeneous spaces: Since H* (BC; k) is a polynomial algebra in the case of Theorem 8.1, it is free as a commutative algebra. One can study a mapping H* (BC; k) —> C - (BC; k) that chooses a representative for each generator. If this mapping is systematically homotopy equivalent to an algebra homommphism, then we have a situation like Cartan's proof of Theorem 8.2 and a chance to prove the collapse result. To study Tor as a functor 'up to homotopy,' we fix the class of differential objects we plan to consider, and choose the bar construction as the basic object; Torr (M, N) = H(13* (M, N)). Let (F) = B . (k, r, k) and recall that

293

8.1. Homogeneous spaces

(B (F), d• + s) is a differential coalgebra where d• is the internal differential ( = -1 ) 1+deg

(and

I

d* (Vy1

•••

I -rn1) =

—1 dr (-Yi) • • • 7n],

[T1 • • • 1

is the external differential, n-1

6 ([eY1 1 • • • 1 7n]) =

'••

'

'••

7n] ,

and the comultiplication is given by A(E-Yi

I

170 = >7 in_o [71

••

I

••

I

7i - 1] 0 [7i • • • 7n].

If f: A —> r is a mapping of differential graded algebras, then f induces a mapping of differential coalgebras, B(f): B (A) —> B (F). The central observation that makes the extension of Tor possible is that maps, B (A) —> B (r), of differential coalgebras carry the chain homotopy information we need. FurtherB (F), is B (f) more, not every map of differential graded coalgebras, B (A) for some algebra mapping, f: A —> F. These observations are made precise in the following result of [Stasheff-Halperin701. Theorem 8.18. Suppose A and r are connected differential graded algebras over afield k. Let DCoalg(B (A), Bp) denote the set of morphisms of differential coalgebras, B(A) B(r). Then DCoalg(B(A), B(F)) is in one-to-one correspondence with the set of sequences of k-linear mappings, (f 1 , f 2 , ) where n times

(1) fi : A ® • • •OA—>Fhasdegreel—n, (2) for all n, fiNr).f n (ai 0 • • • 0 an) n-1

i=l

,

—1 i=1

fi

(al

e

'(di

i=1

fn

(a

(s • • • (s)

• • • 0 F1,_1

dA(ai)

•••®

an )

an )

a i±i

) •

f n—i (ai+i 0 • • •

an ).

PROOF: If F: B (A) B (F) is a morphism of differential coalgebras, then the result follows by direct calculation. Given such a sequence of mappings as described above, define the mapping Bf • : B (A) —> B (F) by B.f . Cal

=

•••

an])

y k=1 S(n,k)

[f i1 (ai (g) • • • 0 aii) • • •

f i k (an-ik-ki 0 • • • 0

an)]

294

8. The Eilenberg -Moore Spectral Sequence II

where S(n, k) is the set { , • . • , I Li = )2 } . Another direct calculation shows that this is a mapping of differential coalgebras. (See the papers of [Clark65] or [Wolf77] for more details.) To see how this theorem relates to 'algebras up to homotopy,' observe that, = 2, the formulas above give

for ri

dB (r) f 2 (a 0 b) — f 2 (d A(a) 0b+a®dA(b)) = f i (a • b) — f 1 (a) • f (b),

that is, f i induces an algebra mapping on homology. In order for f i to induce an associative multiplication, f3 is needed to fill in the appropriate chain homotopies. To quote from [Wolf771, " . f 2 is a chain homotopy measuring how far f i deviates from being multiplicative. Thus, in a sense, f 2 atones for the sins of f' —but adds a few of its own, f3 , in turn, is a chain homotopy of chain homotopies, and atones for the sins of f i and f 2 —but ... and so on." Definition 8.19. A sequence of mappings, (f', f 2 , . . .), that arises from a differential coalgebra morphism B (A) —> B(r) is called an shm (strongly homotopy multiplicative) map, denoted A > F. We also say that a mapping of differential graded modules over k, f: A —> is an shm map, if there is a sequence as above (f i , f 2 , .) with f i = f. The terminology of 'strongly homotopy multiplicative' mappings and the systems of higher homotopies that express the relations implied by associativity were first codified in [Sugawara571 and lStasheff63]. Extend the category DGAlgk of differential graded algebras over k to a new category DASHk with the same objects as DGAlgk but with the sets of morphisms given by DASHk (A, r)

= DCoalg (B (A) , B(F)).

The category DGAlgk embeds in DASHk by sending an algebra homomorphism f: A —> r to the sequence (f, 0, 0, . ). We denote a morphism in DASHk by f: A > To prove Theorem 8.1, we develop the notions of algebras and modules over algebras having sh-structure maps and extend the functor Tor to accept sh-objects and shin maps as variables. This extension was carried out in [Gugenheim-Munkholm74]. [Stasheff-Halperin70] observed that, for the differential graded algebra C* (BC; k), satisfying the assumptions of Theorem 8.1, there is an shm map, H* (BC ; k) > C* (BC ; k), inducing the identity mapping on homology. By getting the sh-module structure correct, the desired isomorphism on Tor follows.

The following series of remarks and results, stated without proofs, gives the steps in this program leading to the proof of the powerful collapse theorem of [Munkholm74]. The interested reader can find details in the references cited along the way.

295

8.1. Homogeneous spaces

A. If r is a differential graded algebra, then a left sh-module over r is a differential graded module, M, together with a sequence of k-linear mappings gn

such that

r 0 • • • r oxf n copies

(1) gn has degree 1 — n, and

(2) the adjoint mappings ad?"' :ro•••or map.

Hom(m, M) form an shm

A similar definition can be given for right sh-modules over F. The definition of sh-modules allows us to induce sh-module structures from mappings in DASHk : If f : A > r is a mapping in DASHk , then r is an sh-module over A via the mapping f. B. If A and r are differential graded algebras, f: A s r, a mapping in DASHk, M, N are, respectively, right and left differential graded modules over A, and M', N' are similarly differential graded modules over r, then we write g: M > M' and h: N > N' for mappings g: B (M, A, k) —> B (M', r, k) and h: B (k, A, N) —> B (k, r, N') that commute with f as comodules over the coalgebras B (A) and B(r). Such mappings induce a natural homomorphism Tor f (g, h) : TorA(M, N) —> Torr (M', N'), extending the classical case. The following generalization of Corollary 7.7 is due to [Gugenheim-Munkholm74]. Theorem 8.20. Suppose f: A > r is a mapping in DASHk and g: M M' and h: N > N' are sh-module mappings. Then we can write f = (f l, f 2 ) • • • )) g = (g° , g l , ) and h = (h° ,121 , .) with f' : A —> r, g0 : M M' and h0 : N N'. If H(f 1 ), H (g°) and H(h°) are all isomorphisms, then so is Tor f (g, h).

C. We say that a differential graded algebra r is she (strongly homotopy commutative) if the multiplication on F, m: F F 1', is an shm map. Examples of such she algebras are C* (X; k), for X a space and multiplication given by the cup-product. When Y is an H-space, then (S2Y; k) is an she algebra (IlSugawara601 and [Clark651). lStasheff-Halperin70] observed an important property of she algebras of a certain type that applies to the problem of computing H'(GIH;k). Proposition 8.21. Suppose r is an shc algebra and H(r) k[x i , , xn]. Then there is an shm map, V): r inducing the identity on homology.

The proof follows by induction on the number of polynomial generators. For n = 1, the definition of an she algebra allows us to construct the mapping

296

8. The Eilenberg -Moore Spectral Sequence II

by sending the generator to a representative of its class. Generally, the choice of a representative leads to a shm map k [xi ] • • • k [x n ] re). • • F that can be multiplied down to an shm map k [xi, , x] because m r(gr,r is slim. To apply the proposition to the problem of homogeneous spaces, there are shm maps H* (BG; k) C* (BC ; k) and H* (BH; k) > C* (BH; k) inducing the identities on homology. The desired collapse theorem can be derived from Theorem 8.20 if the sh-module structures can be brought into place. This requires a careful consideration of when diagrams such as H* (BG; k)

C* (BG; k)

Bel

H* (BH; k)

) C* (BH; k)

commute up to homotopy. D. Proposition 8.21 does not restrict us to the case of homogeneous spaces but applies to the cochain algebras of spaces with H* (X; k) a polynomial algebra. Thus the extra structure of Lie theory is unnecessary for the collapse theorem. The general theorem is due to [Munkholm74]. Theorem 8.22. Suppose we have the pullback diagram with 7, a fibration and B, simply-connected: Ef E ir

X

B.

If H* (E; k), H* (B; k) and H* (X; k) are all polynomial algebras over k mat most countably many variables, and if char k = 2, we suppose further that Sq l H* (E; k) and H* (X ; k), then, as graded k-vector spaces vanisheo H* (E k)

TorH *(B, K) (H* (X; k), H* (E; k)).

That is, the Eilenberg-Moore spectral sequence, converging to H* (E k), collapses at the E2-term.

Theorem 8.1, the collapse result for homogeneous spaces, follows as a corollary. Theorem 8.22 also applies to two-stage Postnikov systems. The restriction on Sq l , the cupi -product, is best possible by the example of [Schochet7 1 1 . [Wolf771 gave another proof of Theorem 8.1, similar to [Munkholm741, but using the geometric properties of homogeneous spaces. [HusemollerMoore-Stasheff741 developed a general theory of differential homological algebra that applies to the problem of computing H,,(G I H; k) via the homology Eilenberg-Moore spectral sequence and they are able to prove a similar collapse result.

297

8.2. Differentials in the Eilenberg-Moore spectral sequence

The methods of strongly homotopy multiplicative maps in differential algebra and more generally, the notion of algebraic properties up to homotopy were developed into powerful organizing principles under the rubric of operads during the 1970s ([May72]) and this idea has found its way into mathematical physics and quantum algebra ([Loday96]), [Stasheff97]). The perturbation theory of a differential on a resolution or of a differential on a graded algebra that underlies Theorem 8.15 was introduced by [Gugenheim60], [Liulevicius63], and [Gugenheim-Milgram70], and has enjoyed considerable development (with an especially neat formalism due to [Brown, R67]). For a survey of these advances, see the papers of [Huebschmann-Kadeishvili91] and [Lambe92]. 8.2 Differentials in the Eilenberg-Moore spectral sequence

The underlying category for the Eilenberg-Moore spectral sequence has objects differential graded algebras and modules over them where the module structures are induced by algebra mappings and multiplications. The functor Tor, defined by resolutions, encodes nontrivial relations in the classical sense of syzygies. These relations are due to the multiplicative structures involved and the interplay between homology and the product. When a pair of products vanish, u • y = O = y • w, a secondary operation, u y, :t0 may be defined that was introduced by [Uehara-Massey571. In this section, we introduce the Massey triple product and its generalizations due to [Massey581, [Kraines661, and [May68]. A motivating example is the loop suspension homomorphism that may be described as follows: The functor k) is representable as [ , K (k, n)] for each n > O. This description of cohomology allows us to define a mapping H' (X; k) —> H n-1 (0X; k) by applying the topological loop functor: ,

: [X , K (k, n)] —> [OX , OK (k, )] = [OX,K(k,n — 1)],

f

0(f).

[Eilenberg-Mac Lane50] introduced the loop suspension homomorphism in their study of the spaces K (II, n) and [Serre511 developed it in homology (§6.2), relating it to the Leray-Serre spectral sequence. [Whitehead, GW551 proved that S2* annihilates products and furthermore, S2* is an isomorphism for n less than three times the connectivity of X. [Kraines661 showed that the higher operations based on products, the higher Massey products, are also annihilated by S2* . These operations were developed further and shown to determine the differentials in the Eilenberg-Moore spectral sequence. [May68] applied these results to obtain collapse theorems similar to those of §8.1. To begin we describe the cohomology loop suspension homomorphism for a fibration and, in particular, for the path-loop fibration. We next introduce the classical Massey triple product and describe some of its geometric applications. Then we consider the generalization of the triple product to an arbitrary number of variables. These higher order operations are related to the cohomology loop suspension homomorphism via the differentials in the Eilenberg-Moore

298

8. The Eilenberg -Moore Spectral Sequence II

spectral sequence. Finally, we consider matric Massey products that generalize the higher order products and a structure theorem that expresses the differentials in the spectral sequence in terms of matric Massey products. The cohomology loop suspension homomorphism

Dual to the transgression homomorphism (§6.2) is the suspension (or loop suspension) associated to a fibration, F —> E —> B. It is defined as an additive

relation by the homomorphisms Hq (B) Hg(B,*) —> Hq(E, f) Hq -1 (F)

as in the diagram: Hq -1 (*)

Hq (B , *)

(B)

Hq (*)

> H_1(F) 6 > Hq (E , F)

Hg (E)

> Hg (F)

The homomorphism 12* : (76)'(im 6) —> Hq' (F)/ker 6 im 6 is induced by 76'. It is a homomorphism from a submodule of Hq (B) to a quotient of Hq (F).

For the path-loop fibration, E = P X *, so 6 is an isomorphism and 0* is a homomorphism Hq (X) —> Hq' (0 X). [Serre51, Ln° 3 1 showed for the path-loop libration, that Sr has the same image as the monomorphism E°–' q -1 (QX) and the same kernel as the surj ection H q (X) , where Eq*'* refers to the Eq -term of the Leray-Serre spectral sequence for S2X P X > X (see Proposition 6.10). In the Eilenberg-Moore spectral sequence the loop suspension homomorphism will be seen to have a very simple expression in the case of the path-loop fibration. The motivating problem for §8.2 and its subsections is to describe the kernel of 12*. Following [Smith, L67 1 , we approach this question by looking at the filtration due to the homological nature of the Eilenberg-Moore spectral sequence. Fix a fibrati on 7r: E B with connected fibre F and letf:X—>B be a continuous mapping. Let Ef —> X denote the pullback fibration. Recall that Ef c X x E. Let —

if:XxE—>XxBxE

and f:XxE—>XxBxE

be given by fr(x, e) = (x, 7r(e), e) and j(x, e) = (x, f (x), e). It follows that ir and f agree on the subset Ef . These data determine a difference homomorphism, (f - fr)* : H* (X x B x E; k) —> H* (X x E, Ef; k), following

299

8.2. Differentials in the Eilenberg-Moore spectral sequence

[Steenrod47], that is defined as a lifting at the cochain level in the diagram: C* (X x E; k)

> C* (E f ; k)

O.

f*—**

\\, C* (X x B x E; k) Fix a field k

and write H* (X) = H* (X; k). By the Kiinneth theorem, we have

H* (X x B x E)

H* (X) 0 H* (B) H* (E).

Next consider the boundary homomorphism : H 1-1 (E f) Hq (X x E) of the long exact sequence for the pair (X x E,E f). Together, ( J it)* and (5 determine an additive relation 4) similar to the suspension and the transgression: (i )* H* (X) g H* (B) H* (E) > H(X * x E,E f) < (E f) -

where 4) : ((f it)* ) -1 (im (5) —> H* (E f) I ker (5 is induced by (f — fr)*. For the path-loop fibration, ev i : P B —> B, we obtain the fibre QB as the pullback over a choice of basepoint 7): —> B. The difference homomorphism reduces to ev; in this case because PB *. Thus the following diagram commutes: evi H* (B, *) ) H* (PB , Q B) -

k

if* (B)

ek

0—evir

) H* (P B, Q B)

In this case, the additive relation 4) is Sr. For a general fibration, recall that H* (X) 0 (B) H* (E) = B -1 (H* (X), H* (B), H* (E)) is the ( —*column of the bar construction for TorH.(B)(H* (X), H* (E)). Let 0:(X)(S) fl* (B)0 H* (E) —> H* (X )0 H* (E) be given by 0(u0a0u) = üa e y + u e ãv, where = (-1) 1 + deg xx. This is the bar differential and so we get an epimorphism T: ker ç5 TorH — ) (H* (X), H* (E)). We next identify ker 0 with the domain of the additive relation 4). Consider the diagram:

H* (X) 0 (B)

H* (E)

H* (X) ® H* (E)

300

8. The Eilenberg -Moore Spectral Sequence II

The left triangles commute by the definitions of f and it and the Kiinneth theorem. The right triangle is exact. It follows that:

(f -

fr)*(w) E im 6 < > j* (f

-

fr)* (w) = 0

< < > 0(w) — O.

Thus we can write 4): ker —> H'i (E f) I ker 6. Proposition 8.23. Suppose that B is a simply-connected space. Then

F° H* (E f) = im{i* : H* (X) ® H* (E) —> H* (E fil

Furthermore, (E f) I F° H* (E f) is additively generated by elements of the form 4)(w) for some w E H* (X) ® (B) Ø H* (E).

PROOF ([Smith, L67]): To prove the assertion about F° , we consider the mapping k C* (B; k) given by a choice of basepoint. Then there are algebraic spectral sequences with E2 -terms related by the map induced by the unit Tor' (H* (X), H* (E)) —> Tor'X (B) (H* (X), H* (E))

and converging to Tork (C* (X), C* (E)) —> H* (E f). However, since k is a field, Tork (H* (X), H* (E)) H* (X) ®k H* (E), concentrated in the 0column. Furthermore, the mapping induced by the basepoint is given by the morphism induced by the inclusion Ef c X x E, as seen by examining the bar constructions. In the Eilenberg-Moore spectral sequence converging to H* (E f), there is an edge phenomenon similar to the one in Example 5.E for the Leray-Serre spectral sequence. Since dr has bidegree (r, 1— r), d r 0 on Er-1, * for r > 2. Thus E* ErT, 1, 7 im d and there is a sequence of epimorphisms ker

T

Tor

(3) (H * ), H * (E))

-1*

»

cc

•

Thus Eoo -1, * = F'H*(Ef)IF°H*(Ef) is generated by the image of the classes from ker under T. Notice, however, that 4) factors through T ker

» Torc i

(C* (X) , C* (E))

H* -1 (Ef ) I ker 6.

As the diagram identifying the domain of 4) with ker 0 shows, both T and 4) have the same kernel and the second assertion is proved. In fact, we can say more. Recall that we have the short exact sequence 0 —> F° H* (Ef)

> F-1 H* (Ef ) —> E00 -1 '* —> O.

8.2. Differentials in the Eilenberg-Moore spectral sequence

301

Following [Smith, L67 1 , there is an explicit way to represent all of the elements in F-1 H*(Ef ) via (1). If w E ker, 0, then we can write w = Ei x 0 a® y. Let Xi, Ai and Yi denote choices of cochain representatives for the classes appearing in w. Since 0(w) = 0, there is a class U E C* (X) C* (E) with do (U) = Ei ±XiAi ±Xi AY. Form the element Z =

X, [A]Y, — U E B . (C* (X), C* (B), (E)).

* * The association of w with the class of Z in Tor C*(B) (C (X) , C* (E)) has an indeterminacy given by F°H*(Ef). We can represent the suspension homomorphism by choosing representatives and varying a class given by (1)(W) over the indeterminacy. The result is well-defined in the spectral sequence. Corollary 8.24. For the path-loop fibration, the image of the loop suspension homomorphism is given by F-1 H* (0B). Furthermore, this homomorphism is given by the rule: x E fi n (B) goes to [X] E Torc iinB) (k, k) for any representative X of the class x.

Finally, we examine the consequences of having a spectral sequence. Since —1 (k , Tor C* (B) k) is a quotient of TorH1. (B)(k, k), we can use the E2-term to help describe the image of S2* . In particular, the bar construction differential B -2 _> B ' is given by the multiplication on H* (B) and so QHn+1 (B) -1* (k k) by way of the inclusion, x Tor H* [x]. We may take the domain ( B) of the loop suspension to be QH*+1 (B). [Whitehead, GW551 showed that S2* annihilates decomposables. The codomain of 12* is Tor (B) (k, k) and is contained in the space of primitives in H* (GB) with respect to the coproduct on the bar construction. We examine the consequences of connectedness for the loop suspension homomorphism in a result due to [Whitehead, GW551 and [Smith, L67 1 . Corollary 8.25. If B is n-connected (n > 1), then : Q Hq+1 (B) —>

Prim Hq (S2B)

is an isomorphism for q < 3n.

PROOF: By Corollary 7.19, it suffices to show that no element in E2-1 ' q for q less than or equal to 3n ± 1 is in the image of a differential, and hence such elements survive to E. Since dr has bidegree (r, 1 — r), an element in E2-1 ' t would be hit by a differential dr only if there are elements in Er-(r+1 ),t+r-1 Since E2 (r+1) "9 = {0} for s < (n+1)(r+1) by the connectedness assumption, dr : Er (r+1)' s —> Er-1, s — r+1 is zero for all r > 1. From s < (n +1)(r + 1)

302

8. The Eilenberg -Moore Spectral Sequence II

we get s — r +1 < n(r + 1) + 2. Starting at r = 2 it follows that t = s — r +1 < 3n + 2. The total degree of such elements is then less than 3n + 1 and so the result follows. These results make the Eilenberg-Moore spectral sequence the tool of choice to study the loop suspension homomorphi sm. We take up the problem of describing possible differentials in subsequent sections. We remark that F'H* (E f ; 1F,) as described by the partial homomorphism .tD is determined by geometric maps and hence is closed under the action of the Steenrod algebra. [Smith, L67 1 asked if the higher filtration pieces were also closed under the action of the Steenrod algebra. This problem is taken up in §8.3. Another interpretation of the filtration submodules in the special case of the Eilenberg-Moore spectral sequence converging to the cohomology of a classifying space was given by [Toomer74] and is related to the LustemikSchnirelmann category of a space. Generalizations of this notion are due to [Fadell-Husseini92] and [Strom]—essential category weight of a cohomology class, and by [Rudyak991 strict category weight. In both cases the EilenbergMoore spectral sequence and Massey products (even matric Massey products) play a role. Massey's triple product

Let (F, d, ,a) denote a differential graded algebra over a ring R and denote the multiplication on r and H(r) = H(r, d) by p(u, y) = u v. Suppose [u], [y] and [w] denote classes in H(r), represented by u G rP, y E Fq and w G F. If [u] • [y] = 0 = [ y] • [w] in H(r), then we introduce a new cohomology class definable because [u] • [v] • [w] = 0 for Iwo different reasons. It is constructed as follows: Since [u] • [y] = 0, there is an element s G rP±q — 1 with ds = g • y (recall that g = (-1) 1 ±cleg n u). Similarly, there is an element t E q-Er - 1 with dt = Pi • w. The element g • w + g. t determines a cocycle in rid+q-Pr — ': d( • w + • t) = (-1)P±gds • w + (-1)13 • dt = (-1)13-"ft • y • w + (-1)P+q+lfi, • y • w =

O.

We define the Massey triple product of [u], [y] and [w] as the set of all cohomology classes that can be defined in this manner: [u], [y], [w]

}[§ • w + g. t] E HP-Eq+r —l (F),

where u, y, w represent

[u] E HP (r), [y] G Hq (r), and [w] G Hr (r), respectively, and s, t vary over choices that satisfy ds = g. y and dt = p. w.}

By regarding our choices more carefully, we can identify the indeterminacy.

8.2. Differentials in the Eilenberg-Moore spectral sequence

303

Proposition 8.26. The Massey triple product ([u], [y], [w]) is an element of the quotient group HP+q+'(F)/([u]• Hq -Er (F) HP -Eq -1 (F) • [w]) .

PROOF: We need to show that different choices do not affect the coset in HP-Eq+' (F) given above. For example, if s and s' are chosen with ds = ft • y = ds' and t is chosen with dt = 17) • w, then (g • w + ft • 1- )

—

(g' • w ft • t)

which, on homology, lies in HP-Eq'(F) • proof.

[w].

—

(g

—

8') • w ,

Similar arguments complete the

Some of the formal properties of the triple product (for example, naturality and relations with the product) are special cases of more general results proved later. Therefore, we postpone listing those properties until then and discuss some of the geometric applications of the triple product. A. If a E 7r(X) and 3 E 7r,(X), then the Whitehead product of a and 13, [a, 3] E p+ - 1(X) is defined as follows: Let f: (IP,ap) (x-,,) and g: (Iq , OP) —> (X, *) represent a and 3, respectively. Then IP x P represents an (p + q)-cell and so a(IP < p)=I x OP U 8l x P SP+q-1 . Define ,*) —> (X, *) given [a, 3] to be the homotopy class of the mapping h: (SP+ by ff (x), if x /P, y h(x,y) = g(y), if x E d/P, y G P. When p=q=p+q-1= 1,[a, 3] = ai3a -1 /3 -1 , the actual commutator, and hence the notation. The elementary properties of the Whitehead product include [a, 3] = (-1)PI [3, a], an indication of the choice of signs for a graded Lie algebra. This operation was introduced by [Whitehead, JHC411 and was conjectured by Weil to satisfy the Jacobi identity for a graded Lie algebra by analogy with the product of [Samelson541 for the homotopy groups of a group-like space. Several approaches to proving the identity appeared about the same time in work of [Whitehead, GW54], [Nakaoka-Toda541, [Hilton551, and [Uehara-Massey57]. [Uehara-Massey571 proved the Jacobi identity by applying the triple product. Consider X = SP V Sq V Sr and suppose a E 7rp (X), /3 E 7rq (X) and ry E ]] eP±q±r-1 . 7r (X) are generators. Form the adjunction space, K = X U[ This is the universal example for the triple Whitehead product. In this case, H* (K) has generators in degrees p, q, r and p + q + r — 1 that we denote by [u], [y], [w] and [z]. Theorem 8.27. For K = (SP V [z] in H* (K) as described,

Sq V S r) U[a,[0, 7]i

([u],[y],[w]) = (-1)P [z],

eP+q+r-1 and [u], [y], [w],

([y],[w],[u]) = (-1)Pq+Pr+P[z]

([w],[u],[y]) = O.

304

8. The Eilenberg -Moore Spectral Sequence II

Thus the triple product is an operation that detects the triple Whitehead product, [a, [0, -y]]. In order to prove that the Whitehead product satisfies the Jacobi identity for graded Lie algebras, form the mapping —1 )Pr [ü,[0, -Y]] + ( -1 )Pg [0, [-Y, + ( -1 )"[-Y, r3]] = [f] and consider the complex K f = (SP ySq ySr )Uf el3+ q±r By the properties of adjunction spaces and Theorem 8.27, [Uehara-Massey571 showed that f is nullhomotopic if the following identity holds for triple products in H* (K f); [

.

( -1 ) P r [u],[u],[11) ] + (-1 ) (1 r [v], [w], [u] + (-1 )" [w],[u], [v ] = O.

This holds, however, by the basic formal properties of triple products and so the desired identity is established. B.The cup product is useful geometrically in the classical theory of intersections of subcomplexes of Rn . If SP and Sq are disjoint spheres in Rn , where n = p q ± 1, then, by Alexander duality, H- (R n — (SP U Sa)) has infinite cyclic generators in dimensions p, q and p q. The cup product of the generators in dimensions p and q is a multiple of the generator in dimension p q and this multiple can be shown to be the linking number of the spheres up to sign (Rolfsen76, p. 132ff.1). If three piecewise unlinked spheres are considered, then a higher order linking number can be introduced by computing the triple product. [Massey68/98] showed that this higher order linking number is nonzero in the case of the `Borromean rings.'

This gives a rigorous proof that, though pairwise unlinked, the three rings cannot be pulled apart. Applications of this higher order link invariant are discussed in the next section along with its generalization to many component links. C. [Viterbo981 has applied the triple Massey product to the problem of determining the minimal number of critical points of a function on a Hilbert manifold X. Through Morse theory this problem can be related to classical LustemikSchnirelmann category and hence to the cup length of a space, that is, the maximal number of cohomology classes with nonzero cup product. Viterbo introduced the notion of tied cohomology classes and tie length to bound the number of critical points. He then used Massey products to give examples of bounds for tie length that are greater than the cup length and hence closer to the desired number of critical points.

8.2. Differentials in the Eilenberg-Moore spectral sequence

305

Higher order Massey products

When two triple products are defined ( [u], [y], [w]) and ([y], [w], [x], then it can happen that there is a choice of elements in (r, d) for which dYi = to • v ft • ti, where dto = ft • v and dti = f, • w dY2 = ti • w + V • t2, where dt2 = ft.) • x.

In this case, 0 element

E

[y], [w])

ft •

and 0

E

to • t2

Y2

qt,], [w], [x]), and we can form the Y1 • X.

This expression is a cocycle and as we vary our choices we obtain a subset of H 1 ' 1 + 1 ' 1 + 1 w 1 + 1 x 1-2 (F, d) that we denote by [u], [y], [w], [x]). A triple product is formed when certain products among three cohomology classes vanish. A fourfold product is defined when certain triple products among four cohomology classes contain the zero cohomology class. In like manner we could define fivefold Massey products and so on. We next give a uniform definition, for all orders, of higher order Massey products. A definition was first given by lMassey581. The following definition (with its choice of signs) is due to [Kraines-Schochet721. Definition 8.28. Suppose (r, d) is a differential graded algebra and [-y1], [ey2], , [yri] are classes in H* (F, d) with [-yi ] E HP' (r, d). A defining system, associated to ([-y1], • • • , [-yri]), is a set of elements (a) for 1 < j <j < n and (i, j) (1, n) with aij E r satisfying (1)

aij c ryi Pri+1+•••+pi i+i ,

(2)

ai , i

-

-

is a representative of [-yi ] in Ifid (F, d),

(3) d(a1) =

-1

E

air • ar+1 ,i .

r=i

To a defining system we associate the cocycle, n-1

al o- ' ar=1,n E 1-1.4±•••+pn—n+2

The n-fold Massey product, ['y,],.. . , [N]), is the set of cohomology classes of cocycles associated to all possible defining systems for ([-y,], . , [yid).

First observe how this definition generalizes the Massey triple product. To give a defining system for ( [u], [y], [w]), we need a "matrix" of values from F, (a ll

(u

a12

U 22 a23 —

a3)

s V t

.

306

8. The Eilenberg-Moore Spectral Sequence II

The defining properties of a Massey product can be summarized in a kind of partial matrix multiplication: s

d

0 ftv 0 vw 0

t =

fa/

t E tï)

ftt gto vw

tv=

The cocycle associated with this defining system appears in the upper right hand corner and represents an element in the Massey product. Extending this matrix notation to more classes recovers Definition 8.28. The n-fold Massey product may be thought of as an 72th order cohomology operation because the data provide that the related (n 1)st order and lower order Massey products must be defined and contain zero in a coherent manner. For example, for ( [-y1 ], [72 ], [ ], [74 ], the defining system provides the boundaries that show 0 E [7211 [73]) and 0 E (VY2], ['y3], [74])• We record some of the properties of the n-fold Massey product. Most of these properties are straightforward consequences of the definition (with the exception of (5)). We encourage the reader to provide proofs of these assertions (one can also refer to the paper of [Kraines661 for details with slightly different signs). —

-y3

Theorem 8.29. Let (F, d) be a differential graded algebra over a ring R and [-y1 ], , [-y.n ] be in H*(r , d). Where the following make sense, they hold: (1) (Linearity) if À E R, then, for all 1 <j

AVM,

c ([71],

A([711, • • • ,

n, [7n])•

(2) (Naturality) if f: F —> A is a morphism of differential graded algebras,

then

f*[-ri], • • • , [N])

C

U.* [71],.

f*ND.

(3) (Associativity) for y E H* (r, d), ([

'y'],...

VTY1], • • •

v C

[N])

TA],

•••

)

[N ]v)

C ( -1 degv (V[71],

[N])

Suppose further that (r, d) has a cuppproduct satisfying the Hirsch formulas (Definition 8.12), then ([71],

r-rni) n ( -1 ) deg u( ['ri

],

, ['yt],v[7t+1],

, ['yn])

307

8.2. Differentials in the Eilenberg-Moore spectral sequence

(4) (Symmetry) if pj = deg -y, and l = PrPs then 1
(n — 1)(n — 2)/2,

bril, • • • , [-nD = (- 1 ) 1 Tril, • • • , [-rn, ]) (5) If k = Fp for p, an odd prime and [-y] E B 2m+i(x;gp) -

-

Pm [y]

e

(

[y ], p

then

• , [7». times

Property (5) plays a role in the analysis of H* (S2X; 7p) when X is an H-space; see Chapter 10 and the paper of [Kane75]. In the special case of coefficients in Fp for p, an odd prime, every odd dimensional class has square zero and so one can try to form the triple product. By using a particular choice of defining system based on the cup i -product, [Kraines66] showed that , [7» (p many), is defined. Furthermore, since we can use some of the data in the defining system iteratively, we can define a cohomology operation, denoted [-y] )P that lies in H27nP + 2 (r, d) when [-y] E H2m+1(r, d) and satisfies ,[7]> (p many). [-YDP c Ty], [Kraines66] showed that the operation [7] [-y»P is equal to — OTnn[-y] with zero indeterminacy. The proof is by universal example. This result was thought to be surprising at the time as it relates an unstable operation, the iterated Massey product, to a stable operation — 131'm. In response to a question posed by [Stasheff68], [Kraines73] applied the Eilenberg-Moore spectral sequence to k stage Postnikov systems to extend property (5) and relate ([1] Pk to —APP k l m • • • Pm [7] for [7] E H2m±i (X; ) and X a space for which there is a mapping satisfying certain conditions, f: X —> Ek, the universal example for the iterated Massey product operation. We return to the question of determining ker Sr with the definition of n-fold Massey products. A connection is made through the following formal lemma, the "staircase argument" for double complexes, which is the argument that led to the discovery of spectral sequences (see Lemme 2 of n° 4 of [Leray451 and the discussion of [Bore198]). In this presentation we follow [KrainesSchochet72].

Lemma 8.30.

Let (24,,,,„ d', d") be a double complex and a l , , as elements in A„,„. Suppose d"a +1 — d' a, = 0 for 1 < r < s — 1 and define a = a l — a2 + • • • + ( 1)s—l a8 . Then da = d'a + d"a = + ( - 1) s—i d as and, in the spectral sequence associated to the double complex, if d"ai = 0, then as survives to E, and (-1) 8 ds ([as ]) = [d'as ]. -

PROOF: First we show that if al and a2 exist as assumed, then d1 is zero on ai and d2 applied to the class determined by ai is the same as the class of d'a2. Finally, if a3 exists as assumed, then [dia2] = 0 and d2 [ai ] = O.

308

8. The Eilenberg-Moore Spectral

Sequence II

By the assumptions, it is clear that al, . . . as all have the same total degree so we can picture the double complex with its filtration as in the picture: 0 d"Î

d//

d'

a2

•

)

a3

Fp

Fp+

)

•

Fp+2

Consider the class al — a2. Note that [al] = [ai — a] in Ez = BP/FP+' and, since d"al = 0 and dial — d"a2 = 0, we have d(ai

—

a2) =

+

—

d'i a2

—

da2 =

—

da2.

In the subquotients d'a2 is a class in the same filtration determined by d(ai — a2) and hence d2 [ai ] = [—d'a2]. Denote total s (As , $ ) by T* ; in the proof of Theorem 2.6, d2 is described explicitly as FpTp_pqncrl (Fp-P2Tp-Pq+1)7_,+1

— Fp+ 2 Tp-pq+1

nd -1 (Fp-p4Tp-

Tp_p q nd -1 (Fp_p i Tp +q+1)+FpTp_pq nd(Fp_ i Tp +q- 1 )

d2

Ep+2,q-1

E9P'q

-

induced by

d

2

IF +3 Tp +q+ ind -1 (Fp +3Tp-p q+ 2)+Fp+ 2Tp-p q-pind(Fp + 1Tp-pq )

Observe that d'a2 lies in Fp+2 Tp±q±i and since d(di a2) = 0, d'a2 is in .L'i +2'q -1 . Also a 2 lies in Fp+C_Fp+q rid- '(Fp+1Tp+q+i) and so ai al a2. If a3 exists as assumed, then a3 lies in Fp +2 Tp+, and we get al — al —a2 a l — a 2 + a3 and

d(ai ) d(ai — a2) d(a i — a2 + a3) = cra3,

which is in Fp+3Tp+ nd- '(Fp+3Tp±q±2 ). So, if a3 exists, d2 [a i ] = 0, that is, ai persists to E3. The general case of s > 3 is similar. We apply this lemma immediately to the algebraic Eilenberg-Moore spectral sequence for computing Tor r (k, k) = H (B • (k, F, k), d) to prove a result of [May68] that was conjectured by Massey. By the results of §7.2, we take the El-term of this spectral sequence to be 13* (k , H (F), k).

8.2. Differentials in the Eilenberg-Moore spectral sequence

Theorem 8.31. If ([ui],

309

,[us ]) is defined in H(r,d), then [ul

••

survives to Es _, and

ds — i([ui I* • • us]) PROOF:

I

tis]

E ±R[114],... l[us])].

Let ((kJ ) be a defining system for ([u i ] • , [us ]) with bi ,, = u. In

B . (k,r, k) define bss ]

a1 = [bil l a2 =

[bll b22 •

ai =

[bli,

as _, =

br —1,r-1 br,r+1 br+2,r+2

I

•••

I

•

bss]

bii_i+1,.]

[bir brd,

where the summation for al is over all sequences 1 < i 1 < i 2 . • < i1_ 1 < s such that j, —1 <1, — s < 1 and i — ii+1 1 < 1. We have d'ai = 0 (since dr (bii ) = 0) and, by a routine calculation, d'ar+i— Sur = 0 for each r (where 6 is the external and d• the internal differential on B . (k, r, k)). By the staircase argument, ds_i [al] = ±[ba 8 _1] =

Since [al] =

[u l • • • us ],

Corollary 8.32. If ['al],

[

r

r•

brsi G ±[(Ui, • • •

lUsA•

the theorem follows. , [us ]) is defined in H* (X; k), then

S2*([u i ], [u2 ],... , [us ])) = O.

PROOF: In the Eilenberg-Moore spectral sequence, converging to H* (AX; k), . . ,[us ]A is an element of and its survival in the spectral sequence determines its image in H*'(QX;k) under S2* by Corollary 8.24. Theorem 8.31, however, shows that [([ud , , [us ]A is a boundary at E8_ 1 and hence it does not persist to E. Thus 1 must annihilate ([ul], . , [us ]). In low dimensions the notion of n-fold Massey products can be used to study links with n components that are unlinked pairwise, three-wise, etc., up to (n — *wise. The class of Brunnian links ([Brunn18921) includes such links as the Borromean rings, and others that are pairwise unlinked, but linked

310

8. The Eilenberg -Moore Spectral Sequence II

in some higher order. Higher order Massey products are ideal for defining invariants of such links and considerable work has been done in this direction (see [Cochran90] and the bibliography there). The classical invariants of knots and links are defined from the fundamental group of the complement in S3 of the knot or link. [Milnor571 introduced the fi -invariants of a link, defined as elements in the quotients of the fundamental group of the complement by its lower central series (Definition §8bis.i5). [Stallings65] conjectured a relation between Milnor's A-invariants and Massey products in the cohomology of the complement. This connection was made precise by [Dwyer75"1, [Turaev761, and [Porter80], and further developed by [Feim-Sjerve841. New link invariants defined by [Cochran90], [Orr91], and [Stein90] are closely related to Massey products and show considerable more subtlety than classical invariants such as the Alexander module. The techniques of Koszul complexes, Massey products and spectral sequences of Eilenberg-Moore type have also found deep applications in the study of local rings. If R is a local ring with maximal ideal, in, then the Poincaré series of the ring R is given by CO

( -1) i dimR/ra (To r( R/ m, R /m))ti .

PR (t) = i

The homological structure of the ring R is closely related to the properties of this series. For example, [Golod62] gave a condition, equivalent to the vanishing of all higher order Massey products in the homology of a Koszul complex, which leads to explicit expressions for PR (t) . IAvramov81 1 introduced a spectral sequence related to the algebraic Eilenberg-Moore spectral sequence that can be used to define obstructions to multiplicative structures on minimal free resolutions of a certain type. In fact, there is a well-defined "dictionary" between ideas in algebraic topology and local ring theory, described in detail by [Avramov-Halperin86]. Another important topological application of Massey products is to the study of Kdhler manifolds. In a fundamental paper [Deligne-Griffiths-MorganSullivan 75 1 proved, using Sullivan's methods of rational homotopy theory, that the vanishing of Massey products of all orders is a necessary condition for the existence of a Kdhler structure. Such a manifold is said to be formal, that is, its real homotopy type is a formal consequence of its real cohomology ring. The relation between formality and the vanishing of Massey products has been studied by ]Halperin-Stasheff791. Massey products also play a key role in the classification of rational homotopy types ([Schlessinger-Stasheff85]) To what degree do higher order Massey products determine differentials in the Eilenberg-Moore spectral sequence? To answer this question and to complete the analysis of ker 12*, we need to introduce a more involved generalization of the Massey products.

8.2. Differentials in the Eilenberg-Moore spectral sequence

Matric Massey products

311

®

One of the features of the homological invariants we have discussed throughout this book is their independence of choice of projective resolution for computation. In particular, we have seen how a carefully constructed resolution can reveal subtle geometric information (for example, in §8.I and §8.2) or permit straightforward algebraic computation (for example, in §7.3). In unpublished work, Massey constructed a version of the bar construction using matrices of elements in the algebras and modules and differentials defined with the slide products of [Mac Lane551. [May68, 69 1 introduced a powerful generalization of the Massey product that was inspired by Massey's bar construction. With this generalization as a tool, May was able to complete the analysis of ker Û* and to obtain some previously inaccessible collapse theorems for homogeneous spaces. Furthermore, a complete conceptual description of the differentials in the Eilenberg-Moore spectral sequence is possible with the matric Massey products of [May68]. In this section, we define May's matric Massey products and discuss some of their applications. We begin with (r, d), a differential graded algebra over a ring R and let Mat (H(1')) denote the set of matrices with entries in H(1', d) = H(F) and similarly, Mat(F), the set of matrices with entries in F. If V = ( 2qi ) is in Mat (r), then we let dl/ = (d-yii) and V = (-T ) = ((-1) 1 +deg "Yij -yii ). For a given matrix V, consider the matrix in Mat ( Z) defined by D(V) = (deg We say that two matrices, V and W, are multipliable if V is a (p x q)-matrix and W is a (q x r)-matrix and the sum eii = deg vik deg wki is independent of k. When V and W are multipliable, we find that VW makes sense and D(VW) = D(V)D(VV) = (e ii ).

We define the matric Massey products inductively. Let V1, V2, , Vri be in Mat (H(0) and suppose Vi and Vi±i are multipliable for i = 1, , n —1. We take in Mat(F) to be a matrix of representatives for the entries in V. When Vil4+ 1 = (0), there are matrices Ai _ l G Mat (F) for which dA i_ i ,i+i = Ai _ i4 Ai , i+i . As in the case of the Massey triple products, Ai _

+

is a matrix of cocycles in F. The set of all associated matrices of homology classes defined in this manner give (Vi, V, Vi+2) a subset of Mat (H(r)). Inductively, we are seeking Aii in Mat (F), for 0 < i < j < n, (i, j) (0, n), so that [Ai_1,i] = Vi in Mat (H(F)) and, for 1 < j — i < n, dA

i -

- 1

k=i+1

-

Aik Aki

It is a straightforward calculation to show that di-4o,, = d( k= il AokAkn) — (0), and so we say that the homology class [Aon ] lies in Vi,... , Vn ). Such

312

8. The Eilenberg -Moore Spectral Sequence II

a system of matrices is a defining system for the matric Massey product ,17„) = {[;40 „] E Mat(H(F)) (Aii ) is a defining system for VI , By reindexing and taking only (1 x 1)-matrices, we recover the definition of the higher order Massey products. Furthermore, the analogues of the properties listed in Theorem 8.29 hold for matric Massey products. We list some applications. A. In a first version of his study of the Eilenberg-Moore spectral sequence, [May68'] used a complex (following Massey) built of matrices and with a differential based on slide cycles to serve as the bar construction. By making precise the homology isomorphism between this complex and the bar construction, it followed directly that the differentials in this algebraic Eilenberg-Moore spectral sequence are expressible as matric Massey products. The details, however, are intricate. [Gugenheim-May74] (see §8.1) reframe this kind of differential homological algebra. Their choice of fundamental object is the 'mapping cylinder' associated to a semi-free resolution and a mapping to a module, M. This leads to a resolution XP

xP*

i'*

H(F)

• • —> X °* H(r)

H(711)

O.

The modules TCP* can be taken to be free over I?, and so one can introduce ordered bases for each XP* and consider each basis as a row vector. In the complex, X Or N, used to compute Torr (M, N), typical elements can be expressed in terms of a row matrix times a column matrix. Given the particular form of a semifree resolution, we focus attention on certain canonically defined matric Massey products that arise from the expression of the differential on X Or N in terms of the bases. It follows that the associated differentials in the Eilenberg-Moore spectral sequence are expressible in terms of these canonical matric Massey products. B. The first immediate corollary of this characterization of the differentials in the Eilenberg-Moore spectral sequence is the collapse theorem of §8.1. For a space with polynomial algebra as cohomology, all matric Massey products vanish, and with them, all potential differentials in the spectral sequence. C. Another immediate corollary is the determination of the elements in ker as those classes in H* (X; k) that are representable as matric Massey products. The proof is similar to the proof of Corollary 8.32. This makes precise the most general notion of when a class is 'decomposable' with regard to the cup product and its associated higher order operations. D. Finally, these homological methods have applications to the dual problem of computing and describing Extr (M,N) (see §9.2). Via these descriptions, [May68] obtained results on the stable homotopy of spheres through the study of ExtA, (Fr Fp) and the Adams spectral sequence. ,

313

8.3. Further structure

8.3 Further structure The Eilenberg-Moore spectral sequence, as developed in Chapter 7, is principally an algebraic tool that relates Torr N) to TorH(r)(H(M), H(N)) for r, M, and N in the appropriate differential categories. The application to topology follows from the second theorem of Eilenberg and Moore (Theorem 7.14) that asserts, for a pullback diagram with 7 a fibration, Ef

E It

X

B,

we have an isomorphism H*(E fj k) Torc.(B;k) (C* (X ; k), C* (E; k)). The spectral sequence converging to H* (E f; k) has been to this point an algebraic artifact of the definition of cohomology. One of the advantages enjoyed by singular cohomology is its rich structure. Cohomology is an algebra via the cup product and Proposition 7.17 shows the homological product on Tor is compatible with that of H* (Ef ; k) and the spectral sequence. When k = 7p , for p, a prime, H* (E f; carries an action of the mod p Steenrod algebra Ap . A natural question to ask is whether this extra structure on H* (E f ; 7p) is also available in the Eilenberg-Moore spectral sequence. In the light of this question, the construction of the Eilenberg-Moore spectral sequence appears somewhat ad hoc and the role of the underlying geometric data in the construction becomes muddled. One route to the existence of further structure would be a geometric construction of the spectral sequence where all differentials and subquotients are seen to be consequences of applying an algebraic functor to a system of spaces and so the extra structure follows for free. We present two geometric constructions of the Eilenberg-Moore spectral sequence, both achieving this goal of uncovering some of the deeper structure. We pay particular attention to the Steenrod algebra structure that can be recovered in this manner. The first construction is due to [Smith, L70] and is motivated by efforts of [Atiyah62], [Landweber66], [Hodgkin68], and [Conner-Smith69] toward proving a Kiinneth theorem for generalized cohomology theories such as Ktheory and bordism theories. The main idea is to develop the properties of the category Top/B of spaces over B in which the product in the category is the pullback to which the Eilenberg-Moore spectral sequence applies. Thus the Kiinneth theorem for spaces, when applied to products in the new category, takes the form of the Eilenberg-Moore spectral sequence. After developing the homotopy theory of the new category Top/B, [Smith, L70 1 constructed a geometric resolution whose associated exact couple gives rise to the desired spectral sequence. )

314

8. The Eilenberg -Moore Spectral Sequence II

In the second construction, due to [Rector70], we introduce a cosimplicial space associated to a pullback diagram. By defining the appropriate face and degeneracy maps on the bar construction, it is easy to see that it is the condensation of a simplicial module. Rector's geometric cobar construction (here presented as a geometric bar construction) takes this idea and derives the Eilenberg-Moore spectral sequence directly from the geometry of the cosimplicial space. This approach is a special case of constructions of IB ousfield-Kan721 to be discussed in the next chapter. Both of the constructions sketched below not only provide us with the sought after further structure but are examples of methods that have been successful in homotopy theory. At the end of this section we mention some further work of [Singer751, [Dwyer80], and [Tumer981 that captures some of the further subtleties of the action of the Steenrod algebra on a spectral sequence. The Eilenberg-Moore spectral sequence as a Kiinneth theorem

For spaces X and Y and a commutative ring R, the Kiinneth theorem allows us to compare H* (X x Y; R) and H* (X; R) OR H* (Y; R). When the flatness assumptions needed for the Kiinneth theorem fail, then the Kiinneth spectral sequence (Theorem 2.20) applies to make this comparison. We now recast these results in a way that allows a generalization. We can write the product X x E in Top as the pullback of the unique mappings X —> * and E *: XxE

* If we apply the Eilenberg-Moore spectral sequence to this diagram, then, over X

a field k, E2

Tork (H* (X; k), H* (E; k)) H* (X ; k) 0k H* (E; k),

and we recover the Kiinneth theorem. In what follows we will take the Kiinneth theorem and the Kiinneth spectral sequence as basic functors on products and develop the geometry of pullbacks to fit this framework. Definition 8.33. Let Top/B denote the category with objects given by continuous functions, f: T f B, where T f denotes the domain off. A morphism in Top/B, a: f —> g is a mapping a: T f —> Tg, so that the following diagram commutes: Tf

a

> Tg

8.3. Further structure

315

This category has been the focus of considerable development since its introduction ([James71, 89 1 and [Crabb-James98]). For the Eilenberg-Moore spectral sequence, the key observation is that the categorical product in Top/B, f xBg, is given by the pullback off:Tf —>B<—Tg:g T(f x

f

1

If

Tg

3

>

B.

This product generalizes the case of B = * to include fibre squares over B. In Top/B the 'one-point space' (a terminal object) is given by the mapping idB : B —> B. Our goal is to prove geometrically the following analogue of the Kiinneth theorem. Theorem 8.34 (the Kiinneth theorem for Top/B). For f, g in Top/B, there is a spectral sequence with E2-term given by E2

TOr H * (id ; k)

(H* (f; k), 1-1-* (g;k))

and converging to H* (f x g; k).

The functor H* (f; k) associates H* (T f: k) to an object f in Top/B. The theorem gives the Eilenberg-Moore spectral sequence when f is a fibration. We get the expected collapse to the Kiinneth theorem when B = *. More generally, it is sometimes possible to substitute other generalized cohomology theories for H*. What makes this approach attractive is an insight of [Atiyah62] that the Kiinneth theorem for K-theory does not follow as in the case of ordinary singular cohomology but requires a construction of spaces whose K-theory has the correct homological properties. A generalization of this sort of construction is needed to obtain the Eilenberg-Moore spectral sequence. In order to develop a homotopy theory for Top/B with the usual geometric constructions, we require a notion of spaces over B with basepoints. Definition 8.35. Let (Top/B) * denote the category of pointed spaces over B, with objects, (f, s), where f: T f —> B is in Top/B and s: B —> T f is a continuous section, that is, f o s = idB. A morphism in (Top/B) *, : (f, s) (g, t), is a mapping, a: T f Tg so that the following diagrams commute Tf

a

B

> Tg

and

Tf

a )

B

Tg

316

8. The Eilenberg-Moore Spectral Sequence II

Tf B is ids, the one-point space (now an initial and terSince B minal object) in (Top/B) 5 is given by (id s , ids ). Notice that H*(tk) is an augmented H* (B; k)-module via the mapping f* and its splitting e. We extend H5 ( ; k) to a functor on (Top/B) 5 by taking the cohomology of the

pair, H' ((f , s); k) = H' (T f ,sB; k).

The geometric constructions to follow are based on the ftmdamental notions of homotopy theory for the category (Top/B).. Proposition 8.36. In the category (Top/B)., the following can be defined: (1) homotopies between morphisms; (2) fibrations and cofibrations;

(3) mapping cylinders, mapping cones, suspensions, smash products, and (4) the Barratt-Puppe sequence.

A homotopy between two morphisms in (Top//3) 5 , say o, i: (f, s)

(g,t),

is a family of mappings, 'Or : (f, s) (g, t) with r E [0, 1] such that 'Or is continuous in r and on T f , and IN = (po and IN = (p i . When f and g are fibrations, this definition coincides with the notion of a fibre homotopy Opanier66, p. 99]). This notion of homotopy gives rise to an equivalence relation. Let [(f, s), (g, t)] denote the set of homotopy classes of morphisms (f , s) (g ,t).

A morphism in (Top/13),„ 0: (f, s) —> (g , t), is a cofibration if it has the homotopy extension property, that is, for any diagram of mappings and a

homotopy ( f , s)

(g ,t)

(h, u)

that commutes when r = 0, there is a homotopy V), : (g, t) for all r G [0, 1], the following diagram commutes:

(h, u) such that,

8.3. Further structure

317

There is an analogous definition for fibrations in (Top/B).. The notions of fibration and cofibration enjoy the usual exactness properties with respect to sets of homotopy classes of mappings. For a: (f, s) (g, t), let (MB (a), smp (,)) denote the mapping cylinder defined by

(m B ( c )) = Tg U (T f x

I)

a(x) (x , 0), if x G Tf t(b) (s(b), r), if b E B, r

where Ms (a)(x) = g(x), if x E Tg, and AI B (a) (y , r) = f (y), if y E Tf and r G I. The section is defined by s m ( a)(b) = t(b) = (8(0, r) for b E B. By forming further quotients by subspaces of (MB (c), s m B ( a)) , one can construct the mapping cone, (CB(a), sc B (a)), of a. The suspension of an object (f, s), denoted by (SB f, SgS), is defined by

T(SBf)= T ic x

(x, 0) (x', 0), (x, 1) (x', 1), if f(x) = f (x' ) , (s(b), r) (s(b), r'), for b G B, r , El,

with SE; f (x, r) = f (x) and S s(b) = (s(b), r). The suspension of a morphism in (Top/B). is also easily defined. We put these objects together to form the Barratt-Puppe sequence of a morphism in (Top/B).: (f, s)

(9,0

(Cs(a), scB(a))

(S f , S Bs) sB(a) (Ssg, S B t)

which gives an exact sequence on application of the functor [—, (h, u)]. The product in Top/B is the pullback. This product can be extended to (Top/B). by giving the section: Given (f, s) and (g, t), then (f >< B 9,s >
T(f AB g) = T (f

B g) I (X, t(b))

(f AB g)(x , g) = f(s) = g(y) and

(8 AB

(s(b), g), (x , g) E T(f x T3 g), t)(b) = s(b) AB t (b).

If we apply a generalized cohomology functor to (Top/B)., then the coefficients of the theory are carried by the analogue of the zero-sphere. In (Top/B). this is the object (8 0 x B, so ) where S° x B is shorthand for the projection mapping S° x B B and so (b) = (1, b). [Smith, L701 gave particular care to identifying the conditions on a generalized cohomology functor for which the desired Kiinneth theorem and spectral sequence results (there are ten conditions). Singular cohomology H* ( ; k) with coefficients in a field k determines one such generalized theory on (Top/B),,. The constructions of Proposition 8.36 allow us to begin to build a geometric homological algebra, following lAtiyah621.

318

8. The Eilenberg-Moore Spectral Sequence II

Lemma 8.37. Given (f, s) in (Top/B)., there is (h, u) in (Top/B). and a morphism (f ,$) —> (h, u) so that (1) H* (a): H* ((h, u); k) —> H* ((f, s); k) is an epimorphism, (2) H* ((h, u); k) is a projective H* ((S ° x B, so); k)-module. PROOF: Let a: T —> (T f sB) x B be the mapping a(x) = (7(x), (x)) where 7r: T f —> T f I sB is the quotient map. If (h,u) denotes the object in (Top/B). with h: Tf IsB x B —> B, the second projection, and u: B T f IsB x B given by b (sBIsB,b), then a determines a mapping a: (f, s) (h, u). Let 13: (T f IsB B)IuB T f IsB be given by 13(x, u) = x. It is well-defined and satisfies 13 o a = id. Therefore, H0 (a) is an epimorphism. Furthermore, by the Kiinneth theorem, H* ((h,u); k) H* (T f I sB x B,uB;k) H* (T f , sB; k) H* (B;k),

and (2) holds.

111

With Lemma 8.37 as a tool, we can iterate the procedure in the proof to obtain a sequence of cofibration sequences: (f, s)

au

> (ho, uo)

> (fi, 8 -1)

(f -1, 8 -1)

(h-1,u-i)

(.f.-2 8- 2)

(f—n1S—n)

(h—n1U—n)

s—n-1)

such that, for all i, (1) H*((h_ i , s_ i);k) is a projective H* ((S° x B, so); k)-module, (2) H0 (a_) is an epimorphism. Such a sequence is called an H0 ( ; k)-display of (f, s). To such a display we associate an exact couple via the Barratt-Puppe sequences. First, we note that when (f, s) is a fibration with section over B, such H0 ( ; k)-displays always exist (Smith, L70, Proposition 3.8]). Furthermore, as required of a resolution in homological algebra, any two displays may be compared with displays that act as free objects in (Top/B). and behave well under H0 ( ; k) ([Smith, L70, Proposition 3.6]). Therefore, we can standardize our displays. For example, if k = Fp, carefully chosen products of Eilenberg-Mac Lane spaces may be taken. Consider the Barratt-Puppe sequence for : (f s _ i ) —> (h_ i ,u_ i ). The cofibration sequences, (f-i, s-i) (f- i - i , s_ i _ i ), in(h_i, u_i) duces mappings, ii: (f_i_i, s_i_i) —> (Ss SBS—i), and so, for each

319

8.3. Further structure

n, we get a geometric filtration of (n f , Ss): (f -711 - n)

(SB;f-n+11 SB 3 -n+1)

Ss An.-

> (•973 f , n s).

SbL -1 A,

Notice that the suspensions are imposed on us by the Barratt-Puppe sequence. However, the cohomology is only affected by a shift in degree by suspension this does not change essentially the module structure over H* ((S ° x B, so); k) or, in the case k = IF, over A.

Theorem 8.38. Suppose B is a connected, simply-connected space. Given (f, s) and (g, t), objects in (Top/B)., with (f, s) a fibration with section over B, there is a spectral sequence with E2

Torfp ((soxs,s030 (If *((g,t); k), H* ((f , s); k)),

and converging to H* (g AB f ,t AB 8; k), where g AB f is the smash product of two objects, g and f, in (Top/B).. SKETCH OF PROOF: We first make the observation that the H* ( ; k)-displays provide partial projective resolutions of H* ((sz f, ns) ; k). By running out the Barratt-Puppe sequences for each line of the display, we can apply H* ( ; k) and splice together the resulting exact sequences; we get an exact sequence

0

H* ((nit° , Sao); k) <— • • •

H* ((n f , Sis); k)

<— • • •

H*((h_ n ,u_ n );k).

Each H* ((h_ i ,u_ i ); k) is a projective module over H* ((S ° x B, so); k) and the effect of S B7n is simply a shift in dimension. Therefore the modules, H* ((SW h_i, Su_); k), remain projective. Apply (g, t) AB - to the display and the other associated spaces. For a fixed positive integer n, we get the space (g AB ST3f,t AB Sis) filtered by successive images: A B f-rilt AB 8-0 —>

(g

AB SBf

tAB SB8_,,,±1) —> • • • —>

(g A13 ST3-1 f_1 ; t AB S73-1 8_1) —> (g AB ST3 f ,t

Ss).

This leads to an exact couple C(n) given by

and

D-1Fq =

H -13+q-- ((g AB Sp n-P f _ p , t AB SB n-PS_p);k)

E–P'q =

H -P-Pq ((g AB S B n-P h_p,t AB S

° u_); k)

320

8. The Eilenberg-Moore Spectral Sequence II

together with a spectral sequence {E r (n), dr (n)}. We make the observation that, for (f, s) and (g, t), objects in (Top/B), (g, t)

AB SB(f,$)

SB((g,t) AB (f,$)),

so El,?, , q (n) Epq+1 (n + 1), when p> 1 — n. Leaving the dependence on n behind, we can define Ef.'q = Epq+n (n),

for p > 1 — n,

and obtain a spectral sequence {E r, dr } associated to (g, t) and (f, s). In order to identify the E2 -term of this spectral sequence, we note that the usual Kiinneth theorem implies that there are natural isomorphisms ((g, t); k)

H . ((so x f3,8 o ),k) If ''((Siz ;"h_i, n'

H* ((g

Furthermore, 0 <—

E2 (n)

((g

AB S731I_i,t AB 5731 U-i);k).

is the homology of the complex

AB SP0,t AB STB1 U0);

H* ((g

k) <—

AB SVh_i,t AB n-1 11_1);

• • • <—

(0

k)

AB h-rt,t AB u_ n ); k) <—

0

It follows that 4'q (n)

Tor H , (( s o x B ,,o);k) (H* ((g,t); k), H* (S73 ( f , s); k).

Accounting for the shifts in dimension, we have, for p > 1 — n, k),

((f s); k )) .

All this was derived from the successive filtrations of (g, t) AB sy3 f , s) with the correction for the dimension shift. A careful argument is needed en route, using the simple connectivity of B, to prove that H3 ((f_, s_ p ); k) = {0} for j < 2p. Thus the spectral sequence converges additively to the desired cohomology. (

In the proof of Theorem 8.38, the homomorphisms in the exact couples and the differentials in the derived spectral sequences are all given by applying H* ( ; k) to continuous functions in the Barratt-Puppe sequence. When k =7p , this means that the spectral sequence is a spectral sequence of modules over the Steenrod algebra A. In order to remove the basepoint from the discussion, one can use the functor ( )± : Top/B (Top/B) * that is analogous to the topological functor that adds a disjoint basepoint—one associates to f, an object in Top/B the pointed object (f+, s) where T(f) = T f LIB, the disjoint union, f + = f on T f and f+ = idB on B, and s(b) = b. With this functor, smash products correspond to regular products in Top/B.

8.3. Further structure

321

Corollary 8.39. Suppose p is a prime, B is simply-connected, H* (B; Tr ) is of finite type and f and g are objects in Top/B. Suppose g is a fibration and H* (Tg; F9p) is also offinite type. Then there is a spectral sequence with

Fr ), H* (Tg;Er)) and converging strongly to H* (T (f x g); 729 ). Furthermore, for r > 2 and E2 TorH .( B;7p )(H*(T f;

for each s < 0, Ers , * is a module over the mod p Steenrod algebra, A r, dr is A r -linear and is isomorphic to (H*(T (f x g); 7p)) as Ar-modules.

g,.

The corollary provides the Eilenberg-Moore spectral sequence with a vertical Steenrod algebra structure that converges to the Ar structure on H* (Ef; Fp). The result answers the question posed by [Smith, L67 1 whether the filtration underlying the Eilenberg-Moore spectral sequence is closed under the action of the Steenrod algebra. It is a more delicate issue to prove that this A r -structure is compatible with the product structure of Corollary 7.18 (for example, that the Cartan formula holds in the spectral sequence). This compatibility does hold, however, and we direct the reader to [Smith, L70, §71 for a proof. This geometric construction also provides an approach to the EilenbergMoore spectral sequence for a class of generalized cohomology theories made precise by [Smith, L70 1 and [Hodgkin681. [Yamaguchi861 has computed h * (S2B) in this way for generalized homology theories and deduced positive results for the Morava K-theory of double loop spaces of complex Stiefel manifolds. [Jeanneret-Osse981 have applied the Eilenberg-Moore spectral sequence to compute the generalized cohomology of fibre squares when the base space of the fibration has h (S2B) an exterior algebra on odd degree generators. This applies to p-compact groups, a generalization of Lie groups suited to homotopy theory methods. The general convergence properties of such spectral sequences is a very delicate issue that remains an open question. Examples due to [Hodgkin751 and [Ghaza189] give indications of the extent to which the Eilenberg-Moore spectral sequence can fail: For example, there are finite {0 } . complexes X for which k* (x) = {0} while k*(S2X) Cosimplicial techniques

In his geometric construction of the Eilenberg-Moore spectral sequence, [Smith, L70 1 introduced H* ( ; k)-displays comprised of sequences of objects in (Top/B) * , {(h_i, u_i)}, whose cohomology algebras are projective modules over H* ((S ° x B, so); k) and that lead to projective resolutions. By analogy with the bar construction, it would be convenient to standardize the construction of spaces that lead to the spectral sequence and furthermore, to base the constructions entirely on the initial datum, the pullback diagram: Ef 7C

X

B.

322

8. The Eilenberg -Moore Spectral Sequence II

[Rector70] carried out this analogy exactly by constructing a geometric cobar construction. When one applies the homology fiinctor, ( ; IF,), to the associated sequence of spaces, one gets an exact couple that gives the homology Eilenberg-Moore spectral sequence with an apparent Steenrod algebra comodule structure. We present this construction for the cohomology spectral sequence. Definition 8.40. A cosimplicial object A in a category C consists of (1) A sequence of objects in C, A° , A l , A2 , .... (2) For each n > 0, a collection of morphisms in C, : A' —> An, for 0 < i < n, called coface maps and (3) for each n > 0, a collection of morphisms in C, : An+ 1 —> An', for 0 < i < n, called codegeneracy maps. The coface and codegene racy maps satisfy the identities:

dj

d' =

di -1 , if i < j,

d os', if i < j, sj o = {id, if i = j or i = j 1, if i > j +1. o =

o , ifi > j.

Examples: A. If V. is a simplicial vector space over a field k, then let Vd.ual be the cosimplicial vector space with (Vd.ual )n = V dual rn . The coface and codegeneracy maps are dual to the face and degeneracy maps of V.. B. Denote the category of sets and functions by Ens. For Y a set, we can apply levelwise the Horn functor, Ens(—, Y) to a simplicial set X. to obtain a cosimplicial set denoted (XY)•= Ens(X„„ Y).

C. lEilenberg-Moore651 introduced the idea of a triple (R,(p,11)), which is a functor R: C C, together with two natural transformations : id —> R and /': R o R —> R satisfying = '00R) and "( R ,P) = id =

(R(P)R =

If S and T are any pair of adjoint functors T S

S:C—>D, T:D—>C,

then the composite functor, ST: D —> D is a triple with the structure maps coming from the adjointness relations. A triple gives rise to a cosimplicial object in C for each object in C, defined by the sequence of functors, R = Rn-1 R and R° = id. Let X be an object in

8.3. Further structure

323

C. Then (CoR(R)X)* (the cosimplicial resolution) is defined by the sequence of objects (CoR(R)X)n = Rn+ 1 X together with the coface and codegeneracy morphisms defined by the natural transformations, for 0 < i, j < n, di — 1r ( pR,—, : Rn —> Rn+ 1 ,

d — Riqp,,,_ ; _i : Rn+' —, Rn .

The relations satisfied by (p and V( imply the cosimplicial identities. Applications of cosimplicial objects derived in this manner to homotopy theory are of considerable importance (see the treatise of [Bousfield-Kan72]). Suppose we have a pair of mappings, f : X —> /3 and 7r: E —> B. To these data we associate a cosimplicial space, G* (X, B, E), defined as follows: Gn(X,B,E)=XxBx•••xBxE ...„.,—, n times

bi, . . . , bn, e), di (x, bi, .. . , bn , e) = { (x,b), . . . ,bi,bi, . . . ,b n , e), ( X,

f(X),

(x, bi,... , bn , 7(e), e),

if i =0, if 1 <j < n, if i = n + 1.

si-1 (x, bt, • • • , bn-kt, e) = (x, bt, • • • , bi—], 6'i, bi+1, • • • , bn+1, e), for 1 < i < n + 1. By the Kiinneth theorem, if k is a field, when we apply H* ( ; k) to each space in Gr* (X, B, E) and let 6 = n (-1) i (di )* , then we get the bar construction, i=o B• (H* (X; k), H* (B; k), H* (E; k)), as desired. We next derive the EilenbergMoore spectral sequence from the cosimplicial space G* (X, B, E). We first normalize the spaces, Gi = Gi (X, B, E) to get a sequence 62 6' 63 I —> G2N —> • • • . * —> G°N —> G N

The normalization goes as follows: Let G°N = G° (X, B, E)10, which is the disjoint union of G° (X, B, E) with a new basepoint, G° (X , B, E) II s. Then define GiN = Gi (X, B , E)/Im u . • • u im di • Since eodi = d 5 +1 od° , the remaining map d° induces a map, Si : and furthermore, bi+1 o bi = s. We next construct something like the H* ( ; k)displays of [Smith, L70 11. Let X° = G°N , and form X' as the mapping cone, .1/8 1, to get a cofibration sequence, X ° —> C —> X 1 . Since 6 2 0 6 1 = >t, we can form the diagram

G2

324

8. The Eilenberg -Moore Spectral Sequence II

where the lift 62 exists by the properties of mapping cones. Inductively, we have XP -1

>*

'JP

6P+1 GP+1

GP

+1 ,/114, = XP

and, from the Barratt-Puppe sequences, we have the sequence of cofibrations GUN

->• X 1 ->•

SXU

GIN

->•

X2

SX I

xl+'

sx 1

GIN

Apply f/*( ; k) to each line of the sequence to obtain an exact couple: =

q (X 1 ; k),

= q(CIA k),

with the maps in the couple given by the cohomology of the maps in the cofibrations and the connecting maps from the Barratt-Puppe sequence. The exact couple in turn gives a spectral sequence. Furthermore, if k = IF, all of the mappings in the couple and hence the differentials in the spectral sequence are Ap -linear. To see that the spectral sequence converges to H* (E f; k), observe that X X E, as a pullback. This mapthere is a natural inclusion E : E f ping satisfies d1 o E = o E and so this induces a geometric augmentation, c: Ef10 G°N with 61 o c = *. Comparing the cofibration sequences for c and 6 1 we get the diagram: Ef /0

G9v

E

El X°

61 )

1

S (E f I 0)

ME

1

)X1

.1) / /7

IS

) SX °.

The condition 6 1 o E = * implies a lift S(Ef /0) —> X 1 that commutes with SE. By induction we can continue through the construction of the Xr to obtain mappings S 1 (Ef /0) —› X / that are compatible with the mappings X/ —>

325

8.3. Further structure

This determines a mapping from the direct limit of the mappings

SX /-1 .

—> D -1 +10-1- associated to the exact couple. Using the techniques of Chapter 3, we get a relation between the target of the spectral sequence and fi* (Ef ; k).

To summarize in the case k = IE, we have a spectral sequence, related to H*(Ef; IF) and, for r > 1, the Er -terms are modules over Ar and the differentials are Ar -linear. In order to establish the analogue of Corollary 8.39 for this construction, we need to determine the E2-term of the spectral sequence. Lemma 8.41. B-* (G1N ; k) H* (G 1 (X, B, E); k)Rim(d 1 )* (1 • • • Tim(d 1 )*) •

PROOF: By induction, we show that Cl (X , B, E) . rn

a

fl* (GI

u•••uimd ;k)

(X, B, E); k)Rim(c1 1-3 )*

e • • •

ED im(di n

The cosimplicial identities imply the assertion for j = O. Since sn o o sn , for i > n, sn induces a mapping G1 (X, B, E) im

such that

or

U • • • U im di

C1

(X,

=

B ' E) /im dri U • • • U im di

= id. This gives us the cofibration sequence

G1–'(X, BE)/imdn U. • •U im d1-1

(X ' B E) I ime+1 U• • •U imd1

G1 (X , B, E) im

U U imd 1 •

To complete the proof, apply H*( - ; k) and the induction hypothesis. Corollary 8.42. The homology of the complex obtained by applying H* ( ; k) to * —>

81 —>

82

9 63

—> • • •

is isomorphic to the homology of the complex H* (G1 (X, B, E); k) with (5 =

E

(_n i (di )*. Therefore the E2-term for the spectral sequence associated t=o to the geometric cobar construction is given by E2

TorH.03 ,0 (H* (X; k), H* (E; k)).

The rest of the proof follows by comparing this geometrically defined spectral sequence with the algebraic bar construction spectral sequence. In the case that 7r: E —> B is a fibration, the second Eilenberg-Moore spectral sequence applies to identify the target of the spectral sequence with H* (E f ; k) when it converges strongly. Finally, if k = Fp, we have derived the Steenrod operations of Corollary 8.39.

326

8. The Eilenberg -Moore Spectral Sequence II

Before leaving the simplicial point of view, we mention other approaches that obtain Steenrod algebra operations in spectral sequences. One way is to construct them directly at the level of the filtered cochains and then keep track of how these operations behave in the spectral sequence. Using the construction of the mod 2 Steenrod algebra operations derived from the diagonal approximation of the Eilenberg-Zilber map (à la [Steenrod531, [Do1d62], or [May70]), this program was carried out, first by [S inger731 for first-quadrant spectral sequences and later by [Dwyer80] for second-quadrant spectral sequences. Singer's work yields not only vertical operations

sq k

Epq-kk

but also a diagonal action Sq k : Er" —> Erk – Q 2 g when k > q. These new operations fill a deficit of operations that should be available in the target cohomology and they also provide more structure on the spectral sequence that can be exploited in computation. (See the papers of [Singer73] and [MimuraMori77] for examples.) Dwyer's work by contrast does not yield unstable operations and creates another deficit. In order for the unstable axioms for the action of the Steenrod algebra to be satisfied by the target of this second quadrant spectral sequence, [Dwyer80] constructs new operations :

E2-P' q ->

2
which interact with the differentials to insure that the unstable axioms hold at E. Analogous operations for odd primes have been constructed by [Mori79] and [Sawka82]. In another direction, the spaces SPX appear as the fibres of path-loop fibrations, and so their cohomology and homology may be computed by the appropriate Eilenberg-Moore spectral sequences. In his Ph.D. thesis, [Hunter891 computed B-,,(ffj +2 Sn+ 1 ; 7 2 ) using this method and further structure. For such iterated loop spaces, however, the mod p homology enjoys the action of the Dyer-La sh of algebra ([ A raki -Kudo561, [Dyer-Lashof621). This structure is particularly useful for the class of infinite loop spaces and it would be useful if the Eilenberg-Moore spectral sequence carried this structure. [Ligaard-Madsen751 and [Bahri83] described an action of the Dyer-Lashof algebra on the homology Eilenberg-Moore spectral sequences, and these results have led to successful computations of the homology of certain classifying spaces found in geometric topology. [Tumer981 has given a uniform construction of operations on spectral sequences beginning from the point of view of [Dold61] and [May701 and generalizing the results of Singer, Dwyer, and Bahri. Chapter 7 and 8 do not come near to exhausting the applications of the Eilenberg-Moore spectral sequence. A particularly nice case of a fibre square is the situation where the spaces X, B and E are H-spaces and the mappings

327

Exercises

f : X —> B and 7r: E —> B are H-maps, that is, if mx and mB are the multiplications on X and B, respectively, then m B o (f x f) f o mx and similarly for 7T. By the properties of pullbacks, the space Ef pulled back from 7r over f, is also an H-space. Therefore, the algebraic input for the Eilenberg-Moore spectral sequence consists of Hopf algebras and Hopf algebra morphisms. Also, the target, H* (E f; k), of the spectral sequence is a Hopf algebra. In the best situation, the spectral sequence is a spectral sequence of Hopf algebras, converging to H* (E f; k) as a Hopf algebra. [Moore-Smith68] exploit this ideal situation to show that, under certain assumptions (satisfied by many two-stage Postnikov systems), the only possibly nontrivial differential in the associated Eilenberg-Moore spectral sequence is dp_ 1 where p = char k 0 and p is odd. This fact has been applied by 1Kane751 and others in the study of H-spaces. In the presence of Hopf algebras, Tor computations are much more accessible. [Cohen-Moore-Neisendorfer79] gave examples of the power of these methods in their study of exponents of homotopy groups. Another source of fibre squares is a Postnikov tower. The (n — 1)connective cover of a space X is defined as the homotopy fibres of the natural mappings to the Postnikov tower: X(rt, • • •

In his Ph.D. thesis [Singer68] used the Eilenberg-Moore spectral sequence to compute H* (BU(2r, co); 7p) and H* (U(2r ± 1, , oc); 7p) where p is a prime and U is the infinite unitary group associated to complex K-theory. Finally, we mention that the Eilenberg-Moore spectral sequence plays an important computational role in the work of [Dror-Farjoun-Smith90] on the cohomology of certain mapping spaces, expression of which is given in terms of the T-functor of [Lannes92].

Exercises 8.1. Prove that the two inclusions i j2: U(n) U(n ± 1) described in §8.1 lead to different homogeneous spaces U(n 1)/U(n). ,

8.2. Prove the fact: Suppose A, B, and C are algebras over a field k. If M is a right A-, right B-module, N is a left A-, right C-module, and L a left A-, left B-module, and Tor/ (M, N) = {0 } = Torrc', (N, L) for n > 0, then Tor6 c (M

N, L) Tor'A eB (M, N

c L)

8.3. Show that, in the case H* (G; k) and H* (H; k) are exterior algebras over k, then Tor RG ; k)(k, H* (BH; k)) is a Poincaré duality algebra. (

8.

328

The Eilenberg-Moore Spectral Sequence II

8.4. U se the Sullivan-de Rham complex for rational cohomology (for definitions and

applications, see the book of [Griffiths Morgan8 11) to obtain the rational analogue of Cartan's result, Theorem 8.2; -

H*(G1H; Q)

Tor s* (BG; Q) (Q, H* (BH; Q).

8.5. Show that a polynomial algebra A on n indeterminates that is also a finitely-

generated module over another polynomial algebra B on n indeterminates, via an algebra homomorphism 0: B A, is free as a B-module (Corollary 3.10 of

Maum681). 8.6. Prove that an E-algebra, A (a graded commutative algebra over a field

k,

generated by a regular sequence), has zero deficiency, df (A) = O. 8.7. Prove the assertions in §8.1 about the values of (Bi)* = H'(Bi;7,) for

: SO(/) SO(n), the usual inclusion and p an odd prime. (Hint: Use the Leray-Serre spectral sequence as needed.) 8.8. Show that the cupi product introduced in the last section of Chapter 5 satisfies

the Hirsch formulas. 8.9. Complete the proof of Theorem 8.18. 8.10. Compute the cohomology of the complement of the Borromean rings. 8.11. Prove the assertions (1) through (4) of Theorem 8.29 for higher order Massey products. 8.12. Suggested by Claude Schochet: Suppose given a pullback fibration of the path-loop libration over a space B with respect to a mapping f: 130 —> B, as in the diagram S2B S2B

1 PB 1 Bo

B.

Show that the Eilenberg-Moore spectral sequence converging to H* (E; k) with E2 TorH , B,k)(H* (B; k), k) collapses at E2 if and only if the Leray-Serre spectral sequence with E2 H* (Bo : H* (S213; k)) converging to H* (E; k) has all differentials arising from transgressions. 8.13. Show that a pair of adjoint functors

TSS:C

D,

T : D C,

leads to a triple and hence cosimplicial resolutions of objects in the category D.

8bis

Nontrivial fundamental groups "One of the advantages of the category of nilpotent spaces over that of simply-connected spaces is that it is closed under certain constructions." E. Dror-Farjoun

The category of simply-connected spaces is blessed with certain features that make homotopy theory tractable. In the first place, there is the Whitehead Theorem (Theorem 4.5) that tells us when a mapping of spaces of the homotopy type of CW-complexes is a homotopy equivalence—the necessary condition that the mapping induces an isomorphism of integral homology groups is also sufficient. Secondly, the Postnikov tower of a simply-connected space is a tower of principal fibrations pulled back via the k-invariants of the space (Theorem 8bis .37). This makes cohomological obstruction theory accessible, if not computable (IBrown, E571, ISch0n90], [Sergeraert941). Furthermore, the system of local coefficients that arises in the description of the E2 -term of the Leray-Serre spectral sequence is simple when the base space of a fibration is simply-connected, and the cohomology Eilenberg-Moore spectral sequence converges strongly for a fibration pulled back from such a fibration. A defect of the category of simply-connected spaces is the fact that certain constructions do not stay in the category. The dishearteningly simple example is the based loop space functor—if (X, xo) is simply-connected, S2 (X, xo) need not be. Furthermore, the graded group-valued functor, the homotopy groups of a space, does not always distinguish distinct homotopy types of spaces that are not simply-connected. A classic example is the pair of spaces X1 = RP2' x S 2n and X2 = S2m x RP 2n; the homotopy groups in each degree k are abstractly isomorphic, 7rk (Xi ) 7rk (X2). If we had principal Postnikov towers, we could use the abstract isomorphisms to try to build a weak homotopy equivalence. However, the cohomology rings over 72 "know" that X 1 is not homotopy equivalent to X2. In this chapter we introduce the larger category of nilpotent spaces. These spaces enjoy some of the best homotopy-theoretic properties of simply-connected spaces, like a Whitehead theorem Oror71]) and reasonable Postnikov towers.

330

8bi'. Nontrivial fundamental groups

Furthermore, this category is closed under many constructions such as the formation of function spaces. Group-theoretic functors, like localization and completion, have topological extensions in this category. A subtler result related to nilpotent spaces is due to [Dwyer74] who showed that the Eilenberg-Moore spectral sequence for the fibre of a fibration converges strongly if and only if the action of 71(B) on Hi (F; A) is nilpotent for all i > 0 (Theorem 8bis .29). In §8 b1s .1, we discuss the various actions of a nontrivial fundamental group. The action of a group on a module leads to a right exact functor, the coinvariants of the group action, whose left-derived functors are the homology groups of a group. This theory is developed briefly in §8bis .2 with an eye to its application to nilpotent spaces. In particular, we construct the Lyndon-Hochschild-Serre spectral sequence associated to a group extension, and the Cartan-Leray spectral sequence associated to a group acting freely and properly on a space. With these tools in place we study the category of nilpotent groups and spaces in § 8 bi s .3. We first prove the generalized Whitehead Theorem of [Dror711. We then discuss the Postnikov tower of a nilpotent space. This tower characterizes such spaces and provides a tool for making new spaces such as the localization of a space à la [Sullivan711. Cosimplicial methods offer a functorial route to localization and we give a short survey of the foundational work of [BousfieldKan721. We also prove Dwyer's convergence theorem for the Eilenberg-Moore spectral sequence. The (co)simplicial constructions described here have proved to be fundamental in homotopy theory. We end with a theorem of [Dror73] that, for connected spaces, any homotopy type can be approximated up to homology equivalence by a tower of nilpotent spaces. §8bis.1 Actions of the fundamental group

We begin with a small digression. Let (C, e, ,a) denote a topological group with identity element e and write g • h for p(g , h). It is an elementary fact that 70(G) is a group with multiplication induced by tt. There is an action of (G) on 7r„ (C, e) defined as follows: If g [g] 70(G) is a point in a path component of G and c: (Sn, e i ) > (G, e) represents a class [a] in (G, e), consider —

g • a • g' : (Sa, el )

(C, e), defined by g • a • g' (x) = g a(x) • g'.

When we vary the choice of g in [g] or the choice of representative for [a], we get homotopic maps and so this recipe determines a pairing 70 (G) x 7r7 (G , e)

77, (G, e),

([g],

[a])

[g a • g'].

Since the multiplication on

70(G) is determined by [g] • [h] = [g • h],we have v„([g] • [h], [a]) = ([9], [h] • [a]).Furthermore, the addition on 7(C , e)

agrees with the operation induced by t and so we have vn ([g], [a] ± [O]) = vn ([g], [a]) + vnQg], PD. Thus 7,2 (G, e) is a module over the group 70(G).

8f". 1. Actions of the fundamental group

331

When (X, xo) is a pointed space of the homotopy type of a countable CWcomplex, [Milnor561 showed how to replace the based loop space 0(X, x 0 ) with a topological group. In this case we write Finx = v,„_1 : 7m(Q(X, x 7rn _i (S(X, xo)) —> 7n-1 (S2(X, xo)) and we have proved the following theorem. Theorem 8'3. Given a connected, pointed space (X, xo), for each n > 1, 7r,„ (X, x0) is a module over 7r, (X, xci ) via the pairing Fin : 7r1 (X, x()) X 7r,„ (X, xo)

> 7r,, (X, x()).

—

Furthermore, when n = 1,

ri (X, x0) acts on itself by conjugation, that is, 01([w], [A]) = [w] • [A] • [w] = [w * A * w —l ] where * denotes composition of

paths.

In our discussion of the failure of the homotopy groups to distinguish the pair of spaces RP 2m x S2n and S2n1 x IRP 2n, we only used the observation that, for all k, .Trk(RP 2m x S 2 n) 71-k (RP2n X S 2m ), as groups. They differ as 71-modules. It is a consequence of the interpretation of the action of the fundamental group as deck transformations that 7r, — Z./2Z. acts nontrivially on the 7Z, factor in 72,„„ (RP 2m X S 2n ) and 72n (RP 2n x S2m) coming from the projective space. Since these factors of E as nontrivial E/ 2E-modules occur in different dimensions, the spaces could not be homotopy equivalent, and the homotopy groups, considered as graded 71-modules, distinguish the spaces as different. The fundamental group acts on other groups when we have a fibration F E L B with connected fibre F. Consider the long exact sequence of homotopy groups: a • • • —> n (F

7rn (E1

e)

a

P.

(B, P(e))

—> 7n-1 (F1 e)

••••

We can induce an action of 7r = 7ri (E,e) on 7,,(B,p(e)) via the composite EB

p* x 1

: 7r X 7rn (B,p(e))

7ri(B,P(e)l X 7rn(B,P(e))

7n( 13 1/3 (0).

Thus we can view 7r, (B, p(e)) as a 7ri (E, e)-module and the mapping p* as a module homomorphism. In fact, more is true. Proposition 8bis.2. When F E L B is a fibration of connected spaces, the long exact sequence on homotopy is a long exact sequence of7r i (E, e)-modules and module homomorphisms.

PROOF: We first construct the action of 7T = taken to be 7,1 _1 ce ). Let es: é1)

here represent [Cl] G

7ri(E,e) on 7rn (F,e)

(F, c)

332

8bi'. Nontrivial fundamental groups

ce ) and c4.) E [w] E 7ri (E, e). [he action of 71- on associates to [w] and [i o a] the class [,3] = [i o a]) , where 13(f1) = w i 0 a(fi) * co -1 . The mapping S2p : S2E —> SIB takes ,3 to po

= (pow)*(poioct())*(pow) -1 = (pow)*cp(e) *(pow) -1

Let h : / x / B denote a pointed homotopy between po0 and cp (o. Construct the homotopy H: Sn-1 x I —> B given by H(fI,t)(r) = h(r,t). Thus H (rt, 0) (r) = (p o w) * cp(e) (p 0 w)' (r ) and H (T1,1)(r) = p(e). Consider the homotopy lifting problem posed by the diagram: S 71— ' x {0 }

'3

(E, ce ) op

Sn—

X/

H

(SIB, cp ( e) ).

Since S2p is a fibration, there is a lifting f-/ : x/ (GE, ce ) with f-I(7, 0)(r) = ,3(7,)(r). Because p o H = H, the mapping ,3' ( 7,)(r) = ( 7, 1)(r) determines a loop in F. We define the action of 7r on 7vn_i (2F,ce ) to be the rule that associates [,3'] to [co] E 71- and [a] E 7vn _i ( F, ce ). By the properties of fibrations this is well-defined and gives a module action. We denote this action by linF' F : 71(E, e) X 7vn_ (QF, ce ) c e ). Notice that i o ,3' ,3. Because the class [,3] represents vnE 1 ( [w], [i o a]) , we have that i* (vnE'F, ( [w], [a]) = vflE i ([w],i* ([a])) and the homomorphism i * : 7rn (S 2F, ce ) —> 7rn _ i (S 2E, ce ) is a homomorphism of 7-modules. Furthermore, if [w] = i * ([wi]) for [ i] E 71 (F, e), then i* (vnF Gwi], [a])) = Finally, we consider the transgression 0: 7rn (B, p( e)) f 7rn _ i (F, e). It is best here to substitute a geometric mapping for 0. Consider the pullback diagram: SIB PB Eevi ,P —

ev

B. B is the path-loop fibration. Because P B is contractible, Here ev : P B Eevi ,P has the homotopy type of F and j* = O. The space Eevi ,p is the subspace of Ex PB given by {(y,)\) y E E, À : I B, such that A(0) = p(e), A(1) = p(y) 1. The mapping j: B—> E ev ,,p is given by j (y) = (e, y). The action of a loop w E [w] E 7r may be expressed at this level as (e, (e, (p o w) 7 * (p o In the definition of the action of 71- on 7,, (S2B, cr( e )), the mapping j* takes the class unn ( [w], [a]) to PnF' El ([0.1] , *([a])) after the identification of the fibre with Eevi ,p . Thus j* = 3 is a 7-homomorphism.

8b's.l.

Actions of the fundamental group

333

After all the relevant identifications are made, we have shown that the long exact sequence • • • —>

(F, e)

13,

(E, e) —> 7,,(B,p(e))

a

e) —> • • •

is an exact sequence of 7-modules and 7-module homomorphisms. In certain cases the 71-action is trivial, even for 7T1 a nontrivial group. For example, if X is an H-space, we have the following result of [Serre511. Corollary 8bis•3• If (X , e) is an H-space, then the action Prx, of 71 (X, e) on (X, e) is trivial for all n.

7rn

PROOF: Let w E [w] E 7ri (X, e). If a: ei) (X, c) represents a class in 7,, (X ,e), then consider the homotopy H: Sn' x —> (9,X ,c,) given by H(t)(r) = ,u((ce *a(17)*ce )(r), h(r, t)) where h: —> X is apointed homotopy between ce and w * ce w -1 . From the definition R ( a, 0)(r) = * a(t1)* ce )(r), e) a(11)(r) and H(7, 1)(r) = (co * a(17) * w')(r). It follows that /4 ([w], [a]) = [a]. This result extends the fact that the fundamental group of an H-space is abelian.

Another rich source of actions of the fundamental group is the notion of bundles of groups (§5.3) over a space. For example, when F E —> B is a fibration, then 71 (B, NO acts on the homology of Fbo , the fibre over bo . In Chapters 5 and 6 the applications of the Leray-Serre spectral sequence involved simple systems of local coefficients, that is, where 71 (B, bo) acts trivially on (Fbo ; R). When the bundle of groups is not simple, then the E2 -term of the spectral sequence need not be a product, even for field coefficients. In the next proposition we give the first case of such a difference. The fuinctor r„2( ) on 7-modules defined in the proposition is the first of a family described fully in Definition 8bis .17. Proposition 81*.4• Suppose g is a bundle of abelian groups over a pointed, pathconnected space (X, so) and Co — G, o . Then Ho(X; g) co /r„2Go, where r2,G0 is the subgroup of G0 generated by all elements of the form [o] • g — g with [a] E 7v = 71 (X, xo) and g E Go. PROOF: Consider the mapping 0: Co —> Ho (X; g) given by g 1-4 g xo. Suppose that a c [a] c 7 1 (X, x0 ). Then we associate to g E Co and a: (A 1 , OA') —> (X, x0) the element g a in Ci (X; g). The boundary of g a is given by the formula in §5.3:

Oh(g 0 a) = h[a"](g) a(1) — g 0 a(0) = (h[al(g) — g) 0 xo•

334

8bi'.

Nontrivial fundamental groups

Since a and g were arbitrary, we see that 0 takes Gco to 0 and so induces a homomorphism : Go /172,G0 —> Ho (X; g). To show that 0 is an isomorphism, we describe its inverse. Suppose u E Ho(X; g). We can write u=

gi 0 xi + Bo(X; g).

Let Ai : [0, 1] —> X denote a path in X joining xo to xi . Then we have the isomorphism h [Ai]: G, Go for each i. We define V): Ho (X; g) —> Go /r,2 Go by [Ad(gi). gi xi + Bo (X; g)) = ( i-ih To see that //) is well-defined, we notice that 'zi) takes boundaries to zero—let h[a 1 ](g) a(1) — g 0 a (0) denote a generator of Bo (X; g). The homomorphism qi) applied to such an element gives h[01 ](h[a- 1(g)) — h[,30] (g) where /3i is a path in X starting at xo and ending at a(i). There is a loop based at xo given by )30 * a * 0,-1 and h[00 * a *

(h[01](h[ct-1 ](g))

—

h[N(g))

= h[00 ](g) — h[130 * a *

which is an element of F„2 Go. Since F„2 Go is closed under the action of 71 (X, x 0 ), qi) takes Bo (X; g) to zero. We must account for the choices made in the construction of If Ai and pi are paths joining xo to xi, then we compare h[Aii(gi ) and hLuii(gi), the images of gi (s, x i with respect to the different paths. The difference between these values can be rewritten h[A](gi )- 41090= htai * Ailh[Ai */./n(h[A](gi )= h[mi * (h[Ai

which is an element in F„2 Go . Thus 1,b is well-defined and the inverse of 0. By Theorem 5.1, it follows for an arbitrary fibration with path-connected base and connected fibre, F E —> B, that the associated Leray-Serre spectral sequence has * H* ( F; k ) „2 H* (F; k) as the leftmost column of the E2 term. Though this seems to put us in murkier waters, we can still see our way to deep results in homotopy theory by studying abstractly the action of groups on abelian groups.

Ea

§82 Homology of groups

If 7r denotes a group and M a module over a ring R, then we say that acts on M, or M is a 7-module, if there is a homomorphism p: it —> AutR(M), where AutR(M) is the group of R-linear isomorphisms of M to 7T

8bis.2. Homology of groups

335

itself. More generally, 7 acts on a group G if there is a homomorphism p: r -> Aut (G). To study modules over a group it we introduce the homology of groups, a homological functor analogous to Tor. The homology groups of a group 7r with coefficients in a 7-module M satisfy the axioms for the left-derived functors (rartan-Eilenberg561) of the functor that associates to a 7-module M its coinvariants,

( ), : m

= m/r2,m.

Here FM is the submodule of M generated by elements of the form am — m where a E it and m E M . (More generally, if G is a nonabelian group acted on by 7, let I G be the normal subgroup generated by elements of the form (ag)g' .) Since b(am—m) = (bab')bm—bm, r2,114- is ait-module. Thus a 7 equivariant module homomorphism induces a homomorphism on coinvariants. The induced action of 7 on M„ is trivial. In fact, M„ may be characterized as the largest quotient of M on which it acts trivially. By 'largest' quotient we mean that, if M/M' is another quotient of M by a 7-submodule M' and 7r acts trivially on If/M', then FM c M' and there is an epimorphism -

m/Gm

>

To see that the functor ( )„ is right exact, we give another expression for M„ when M is a 7-module. Lemma 85. Let denote the ring of integers, taken as a trivial right 7-module. If M is a 7-module, then M„ `z7,0E„ M, where Z7 denotes the integral group ring of 7.

PROOF: Recall that Z oz„ M is defined as the cokernel in the sequence 0 Z7 0 M

1Poizied,

> Z

M —> 0z„ M

0,

where (/) : Zo Z7 —> Z is the trivial right action of 7 on Z, and qp : Z70M —> M is the left action of 7r on M. In this case, 0 = eE and so the sequence becomes E70M

where n (a

m) = m — am.

—

>/14

—

>ZeE„M

—

> 0,

Thus Z oz , M = imn

= m/r„2 .m.

The lemma implies that ( )„ is a right exact functor since Z oz„ ( ) is right exact. Definition 86. The homology of a group 7r with coefficients in a (left) module M Is defined by Hi (7, M) = Tori

M).

We write Hi (7) for Hi (7r, Z) when is the trivial left 7-module.

7-

336

8bi'.

Nontrivial fundamental groups

To compute group homology we introduce a convenient functorial resolution. Let 11„ denote the free abelian group on (n + 1)-tuples of elements of 71 . Then B, is a right 7-module with the 7r-diagonal action (So. 7.. , x n )a = (xoct,

,x n a).

As a Z7-module, 13„ is free on those (n + 1)-tuples with last entry 1. Consider the complex —> • • • —>

•••

E7r

B2 —> Bi

Z7,

>

—

0

where 0= Ei (-1)idi and di(xo, .. • ,Sn) = (So, • • • , oFi, • • • ,x,n): E is the usual group ring augmentation given by E (Ei niai ) = E i ni . This complex is exact since there is the contracting homotopy defined by s(x0 7 . . xn) = (1, xo, ,oc n ) and s(1) = 1. To make the free Z7-module structure evident, we introduce the bar notation for generators over E7F: For n = 0, [1 = 1, and for n > 0, [Si 1 ' ' ' 1 Xn] = (X1S2 ' " Xn, X2X3 ' " Sn, — , 5 n-15n, Sn,

The differential 0 can be rewritten as a= [x2 1 • - • 1 xn], diUxi 1 • • • 1 Sri ]) = { [xi 1 • • • 1 Si-1

1) E B.

(-1)d where for i =0,

1 SiXi+1

" ' X

rd• for 0 <j < n, for i = n.

[X1 1 ' • • 1 Xn-l]Xn,

This is the familiar bar construction on 7F that gives a functorial free right Z7-module resolution of the trivial module Z. Although it is large, the bar construction can still be used to prove structure results. Proposition 8bis.7. If r is a finite group of order 171 and 111 is a 7-module, then every element of M) for i > 0 has order a divisor of 17F1. B„±i , given on PROOF: Consider the 7-module homomorphism , s: B, generators by s( [xi • • • Sn]) = Y E 7r[Y xi • • x n]. We show that

0os±soa=171id: I.••

I x0-9( (-1)idi([Y

E

=171 xi I.••

I

VEIT xi

xrd—

) ± • • • ± ( ±1 i+1 [Y X1

[Ylacil.•• IS•n]) S n])

••

[Yx1 ••

••

Xn]

5 iXi+1

+ • • • ± ( ±1 ) +i [y X1 • • • X n _i]Xn = 1 7 1[5 1 • • • X n ] - S 0 3([S1

••

Xn ])'

• • • Sn]

X1

2

I x3

xn]

8bis.2.

We have used the fact that

E

Homology of groups

[yxi

yEir

E•Exii

I

•••

aos(

I

x2

I

•••

I

337

xn ] = E [y yEr

I x2 • • •

x n ].

If

I xn i ] 0 m represents a homology class in Hn (7r, M), then •

[x i 1—Ixrc]orni)=H

[x 1 3 • • •

x na i ® m 3

and so 1.-K1 times any homology class in Hn (7r, M) is zero. Particular groups may have smaller resolutions. For 7 = E/372,1, the cyclic group of order m, there is a very small resolution:Let t denote a generator of 7, thought of as a multiplicative group. Let IV, denote the free 2 7r-module on a single generator w,. There is an acyclic complex W2n

where T

W271-1

-1472

W1

:Fz

(trace), and 1\r (w2n ) = 3-1-, 2,1±tw2n — l+ (norm). This resolution allows us to compute Hi(E/na) immediately from the complex W. 0z, 17 which takes the form W2n-1) = trW2n-2 — W2n-2

•••±

xm

O

2E --4 27,

xm

O

27, --4

xm

Thus 1/2i+1(17/m7, ) = {0} and H2i (Z/7722E) = 2,1771,E for all i > O. We close this discussion with a useful lemma. Lemma 88. If M is a free left Dr-module, then Hi(7r, M) = {0} for i > 0 and Ho (7r, M) = M„. If M is a trivial left Dr-module that is free over E, then Hi (7r, .111) Hi (7r) 0 M.

PROOF: The assertion about a free module follows simply from the properties of Tor/,,' (17 , M). For trivial modules we can write B. 0z, M = (B.)„ M and the result follows. The Cartan-Leray spectral sequence

Suppose (X, xo) is a connected, pointed space on which a group 7i acts freely and properly, that is, (1) for all x E X, the subgroup Gx = {g c G gx = x} is trivial; (2) every point x E X has a neighborhood U such that gU n U = 0 for all g E 7r, g = 1. For example, if (X, x0) is a connected, locally simply-connected space, then the fundamental group, 7r = 7r1 (X, xo) acts freely and properly on X, the universal covering space of X. Suppose 7 acts freely and properly on X. For any abelian group G, it is a classical result that C.(X17; G) C.(X;G)„, where C.( ; G) denotes the

8bi'. Nontrivial fundamental groups

338

singular chains with coefficients in G and C. (X; G) is a 7-module by viewing each a E iras a mapping a: X —> X. Moreover, Ci (X) = (X; Z) is a free Z7-module for all i. Let F. 0 denote a free, right, 27-module resolution of Z as a trivial 7-module. Consider the double complex C., given by Cp,g =

F2 0a, Cg (X),

OF e 1+ (-1 )q 1 0 ax.

When we filter C.,. row-wise, we get .E.° = F. z.7, Cp (X), d° = 3F g 1, which is the complex computing H.(7 ,Cp (X)). However, c'2 (X) is a free Z7-module and so the E' -term is concentrated in the 0-column where we find Ho (7r, Cp (X)) = C p (X) = C p (X / 7r) . By the appropriate version of Theorem 2.15, cil = 1 = axi„ and so the spectral sequence collapses at E2 to H. (X/7). When we filter by C., column-wise, we get E F2 0,,z,„ d° = 1 e ax. Viewing F2 as an extended module of the form F2 = Ap it with Ap a free abelian group, we get the identification F2 es, C(X) = A 2 0 C(X) = Ap ®11,,(X) = Fp 07,„ 1I (X). This is the complex computing II. (7r, II (X)) and so we have proved the following result of [Cartan-Leray49].

and so

Theorem 8 1'.9 (the Cartan-Leray spectral sequence). If X is a connected space on which the group 7r acts freely and properly, then there is a spectral sequence of first quadrant, homological type, with E22

Hp (77, H,(X))

and converging strongly to II.(X 17r).

More generally, we can use homology with coefficients in an abelian group G taken as a trivial 7-module. For the case of the fundamental group acting on

the universal cover, the spectral sequence has gq H2 (7 r, Hg (X)), where 7r = 71 (X, x0) and converges to IL, (X). From this spectral sequence we prove a theorem that relates the fundamental group and its homology groups to the higher homology and homotopy groups of a space. The case of n = 1 was first proved by [Hopf42] using other methods. This paper launched the study of the homology and cohomology of groups. Theorem 8.1 O. Suppose (X, x0) is a connected, locally simply-connected space whose universal covering space X is n-connected. Let 77 denote 71(X, xo). Then there are isomorphisms Hi (X) Hi (7) for 1 < i < n, and an exact sequence H+2(X) —> H+2(it)

(-Tin+1(5())7r —> H+1(X)

H + ,(7) —> 0.

8'.2. Homology of groups

339

PROOF: Since 111(X) = {0} for 1 < i < n, we have a big hole in the Cartan-Leray spectral sequence converging to H (X) . The theorem follows from interpreting the lowest degree information. The isomorphisms Hi (X) = H C -I Hi (7) for 1 < i < n follow because there are no differentials involved. The first place there is a possible differential is dri+1 4+2,0 —) E62,n+i . This leads to the following short exact sequences: H2 (X) 4+2,0 cr+ 1 —> En7+2,0

—> 0

H71+ 2 ( 7) —› (Hn +1 (X) )1r

O

H +1 (X)

E

1,0

O.

Splicing these sequences together (as in Example 5.D) and substituting H for En2 +1,0 = E 1,0 we get the desired exact sequence. Since 7r1 (X) 7ri(.k) for j > 2, we can substitute 7v,±1(X) for the term in the middle of the exact sequence of Theorem 8bis .10 by the Hurewiez theorem. There is a natural epimorphism of a 7-module onto its coinvariants, so we can truncate the short exact sequence to obtain another exact sequence 7,2 +1 (X) —>

H +1 (X)

—> H

i (7v) —> O.

When n = 1, there is no restriction on X except that it have a universal covering space. We conclude from the theorem that H1 (X) H1 (7) and so Hi (7) 7/ [7, 7] follows from Poincard's classical isomorphism. We also get the short exact sequence of Hopf: 72(X) —>

H2(X)

—> H2 (7r) —>

O.

For an aspherical space, that is, a space X whose universal cover has trivial higher homotopy groups, the integral homology of X is determined by its fundamental group, H1 (X) H 1 (71 (X , z 0 )) for all i. Examples of aspherical spaces are the Eilenberg-Mac Lane spaces K(7v , 1). The study of the homology of groups was one of the motivations for [Eilenberg-Mac Lane53] to introduce the spaces K (7, n). [Kan-Thurston761reversed the process of studying groups using spaces by showing that for any pointed, connected space (X, 0) there X that induces an integral is a group G X and a mapping tx : K(GX, 1) homology isomorphism. Using the same filtration that leads to the Cartan-Leray spectral sequence, we can investigate systems of local coefficients further. In particular, for X a connected space and g a system of groups on X, we restrict our attention to the reduced homology. Consider the group of the reduced chains 0, (X; g) given by sums of expressions g u where u : (A", (A)") —> (X, x0) and g E C O 3 where the singular simplex u sends all its vertices to the basepoint xo, and g

340

8bi'.

Nontrivial fundamental groups

lies in the group over the basepoint. By the properties of homology groups with coefficients, the study of H,(X; g) may be carried out using 0,.(x; g). We quote here a result of [Eilenberg471 and refer the reader to [Whitehead, GW78, VI.31 for a proof.

rm.

Theorem If (X, xo) is a pointed, path-connected space and g a bundle of groups over X, then Hg (X; g) is isomorphic to the homology of the complex G00„C.(X) where Co = G 0 , r = 7r1 (X, x0) and X is the universal covering space of X together with its action of 7r.

The proof follows by a direct comparison. We note a useful corollary. Since iC is one-connected, the complex C. (iC) up to degree two is acyclic and free over 7. Thus we could use this complex as part of a free resolution of over 7 and so identify Hi (x ; g) with Hl (71 (X); G0). This extends Proposition 8bis .4 and the identification of homology groups with local coefficients to include g = 1. The homology of the complex Co ® c.(fc) was termed the equivariant homology of X by lEilenberg471 and it represents one of the basic functors in the study of equivariant homotopy theory. The Lyndon-Hochschild-Serre spectral sequence

The Leray-Serre spectral sequence expresses the relation between the total space of a fibration and its base and fibre. In the realm of groups, a "fibration" is an extension of the form 1

H

7 >

C

1,

where H is normal in 77 and Q"=' 7/H. The "total space" 7 is the extension of the "base" Q and "fibre" H. There is a corresponding spectral sequence relating the homology of a group to a normal subgroup and associated quotient. To describe the spectral sequence we observe that the quotient group Q acts on the homology of H. Let F. —> z7, > 0 be a free right E7r-module resolution of E. Then it is also a free right EH-module resolution. If M is a left 7-module, it likewise is a left H-module by restriction. If g E 7, then define (g).: F. Oar' M – F. (Sax M by (g).(x ® m) = xg -1 0 gm. It follows formally that (g),, commutes with the differential on F. 07ZH- M and so (g). induces a homomorphism (g).: H(H,M) H„(H, M). From the definition of the tensor product over EH, (h). = id for all h G H and so this action of 7r on 1/* (H, M) induces an action of 71H on Hs (H,M). With this bit of structure in place we construct a spectral sequence associated to an extension of groups. —

8bis.2.

Homology of groups

341

Theorem 8 bis•12 (the Lyndon-Hochschild-Serre spectral sequence). Let 1 —> H —> Q 1 be a group extension. Suppose M is a module over 7. Then there is a first quadrant spectral sequence with E2p , q

H1 (Q H ( 1(H M))

and converging strongly to H,,(7, M). PROOF: We begin with some elementary algebraic observations. When we take the coinvariants of a 7-module M with respect to the action of H, we make the action of H trivial on /14H. Thus 7r/H acts on MH by gH • (m+0,m) = gm +III M, where OH M is the submodule of M generated by elements hm—m for h E H and m E M. We next consider the coinvariants of MH- via the Q = it/H action and prove the following formula:

(mH)Q

mir.

This follows from the relation EQ 0 M MH by using the isomorphism o M OQ ZQ oz,, M. Consider the mapping 0 : .11/./H ZQ Oz„ M given by m 10m. Since 10m = 1 ® h —l hm = 1 • h hm = 10 hm, we see that is defined on MH . An inverse to 0 is the mapping L': gH 0m 1—> gm + F J M. That '0 is well-defined follows from the fact that MH is a Q-module. We put these elementary observations to work and suppose that F. —> —> 0 is a free, right, 77-module resolution of Z. Consider the complex computing H.(7r,, M), that is, F. Oz„ M. Since F. Oz M is a E7-module by the diagonal action (g (x 0 m) = xg -1 gm), it is a simple exercise to show that (F. oz M)„ F. 0E, M,

and so F. 0,. M ((F. o M)H)Q • We also have that H(F. Oaf' M) = H.(H, M). We want to compute the coinvariants of the action of Q on these homology groups. Let P. 2 '7 0 be a free right EQ-module resolution of Z. Form the double complex =

Ow (E, o M) H ,

where Q acts on (F 0 .11/f)H by the diagonal action. When we filter columnwise, we get the complex (F. (37.,Q (Fq Oar/ M), d 0 1). However, treating Fq as an extended 77-module (on the left via the antiautomorphism g 1—, g 77), we can write F, = it g A, for some free 7-module A,. It follows that Fq OEH M =

OT,H (Fq O M) =

OH (E7r 0 A, 0 M) = Q 0 A, (s) M,

342

8bi'.

Nontrivial fundamental groups

because (E7r)H = EQ. Thus we have .E.° , q 1. OW 7,7 Q0A g 0M and so Hp (Q, (Fg 0zH M)) = {0} if p> 0 and if p = 0 EL is isomorphic to (Fg earl M)Q = (Fg gair M). Thus = H.(7r,M), and the double complex has total homology given by H. (7r, M). When we filter row-wise, we have =F® (F. 0.- M) and d° = 1 0 d. We argue similarly and write Er as an extended right EQmodule, EQ, to get (F. e,Lii m), do = 1 ® d and so A, 0 H* (H, M) 1 (3„:, H* (H,11/1) and di- = dO 1. Thus E,q

I/2)(Q, ffq(H, M))The first calculations of some of the relations implied by this spectral sequence appeared in Chicago Ph.D. thesis of [Lyndon481 without the benefit of the structure made apparent in the work of [Leray461. The cohomology version of the spectral sequence of Theorem 8bis .12 first appeared in the Comptes Rendues note of [Serre501. [Hochschild-Serre531introduced the spectral sequence of Theorem 8bis.12 along with a formalism that allowed analogous constructions for Lie algebras. Their work was based on a different filtration on the cochain complex whose homology gives the cohomology of a group. Another point of view that gives rise to this spectral sequence is due to [Grothendieck571. The composition of functors spectral sequence (see Chapter 11) for the composition of the coinvariants of the H action followed by the coinvariants of the 7r/H action gives another construction of the Lyndon-Hochschild-Serre spectral sequence. The fact that these alternative constructions give isomorphic spectral sequences from the E2-terni is due to [Bey181]. A simple example of the use of the spectral sequence is the case of the extension associated to a Sylow p-subgroup when it is normal in a finite group. Proposition 81*•13• Suppose

7

is a finite group and P is a normal Sylow p-

subgroup. Then

H(7,7) where

7T

acts on Fp trivially and Q = 71P.

PROOF: The spectral sequence associated to the extension 1 P —> r —> Q 1 has E2 -term given by Ep2 Hp (Q,Hg (P, )). Since the order of Q is relatively prime to p and p times any element in Hq (P, Fp ) is zero, we conclude from Proposition 8bis .7 that Ep2 , qf {0} for p > O. Thus the spectral sequence collapses to a single column given by Er02,g Hg (P, Fp )Q and the result follows. We next consider the "lower left-hand corner" of the spectral sequence. Theorems 8111 '.14 and O s .16 were first obtained by [Stallings651 and [Stammbach661.

8.2. Homology of groups

343

Theorem 8bis•14• To an extension of groups 1 —> H —> x —> Q —> 1, there is an exact sequence 112(7v)

> H2 (Q)

—

>

111[7,

—

(7)

>

—

(Q) ->

0.

PROOF: This is simply the five-term exact sequence from splicing together the differential d2 with the associated graded filtration for H1 (7r) (Example 1.1) as in the diagram: 0

H2(7) 0 _, E

t

d2r

E 2 ,0

> E'02, 1

E6°1

-

>. 0

11

1 0

112(Q)

( 111( 1-1- ))Q

Hi (7)

t E,70

Hi (Q)

t 0

To complete the proof we make the identification (H1 (H))Q H 1[7 , H]. Recall that H1 (H) = H 1[H, H] and that Q acts on this group by conjugation. The conjugation action of 7 on itself has coinvariants 7r/ [7r, 7r] = H1 (7r) and it is the residue of this action that remains as Q acts on H1 (H). The Qcoinvariants of the conjugation action on HI[H,H] are given by 111[7r,H] and so the theorem is proved. 111 To extend this theorem further we recall the lower central series of a group a version of this series will play an important role in the study of nilpotent groups and spaces. Definition 8his• 15• Let

7 denote a group. The lower central series of ir is the family of subgroups defined inductively; • • • C r r 7r C Fr-17v C • • • C r 2 7v C

r27 =

rr7r

7],

To the lower central series of v we associate the completion of 7r: Fr

= 11m 7r/r r 7r. <-r

We can extend the lower central series to higher ordinals by letting 1-'7r = n /3
[7r, 1'07r] when a = /3 + 1 and 1 a7r =

8 Nontrivial fundamental groups

344

Theorem 8bis.16. Suppose 0: 7r —> 7r' is a homomorphism of groups that induces an isomorphism H1 (0): H1 (7) —> H1 (7') and a surjection H2 (0): 112(7v) ,4 112(7'). Then the induced mapping 0: 7r/F r —> 7ri /F r 7r' is an isomorphism for all positive integers r. Furthermore, 0 induces an isomorphism > of completions : —

PROOF: We work by induction. We let V-7 = 7 and F 1 7r/ = 7r' , and so the induction begins trivially. Next consider the extension 1 —> Fry 7r —> 7r/F r 7r —> 1 and similarly for 7'. By Theorem Os .14 we have short exact

sequences H2(7)

H2 (7r/r7r) —* 7r/ [7r, F r 7r] —* H1 (7r)

Hi (7r/F r 7r) —+0

(is* ' H2(70 –> H2 (7ri /r7r 1 ) 1' 7' 77 [7r1 , 7r1 ] —>

(71 ) —>

(7r1 /F r 7r1 ) —> 0

The induction hypothesis together with the Five-lemma implies that 0 induces

an isomorphism Fr 7r/ [7r, Fr7r] rr7r/ [7,

= Fr 7r/F +1 7r, and

Fr7r1. Furthermore, by definition, so there is a short exact sequence

Fr7r7

1 —> 7/F+ 1 7 —> 7r/Fr+1 7r —> 7v/r7r —> 1

and similarly for 7'. The homomorphism 0 induces a morphism of short exact sequences and the Five-lemma implies that 0 induces an isomorphism of 7r/F +1 7r

with 7r7F +1 7r / .

Applications of the Lyndon-Hochschild-Serre spectral sequence abound. We refer the reader to the book-length treatments of the cohomology of groups by [Brown, K82 1 , [Evens93], [Weibe194], [Thomas86], [Benson91], and [AdemMilgram94] for more of the algebraic details, applications, and structure of this spectral sequence. [Huebschmann81, 91 1 has studied the differentials in the cohomology Lyndon-Hochschild-Serre spectral sequence and made successful use of this information to compute the cohomology of some important classes of groups. Some of the exercises at the end of the chapter expose the algebraic possibilities afforded by this tool.

Nilpotent spaces and groups When a group fails to be abelian, the lower central series is a composition series whose consecutive quotients are abelian. One direction to generalize the category of abelian groups is to consider groups whose lower central series is bounded in length. A group 7 is said to be nilpotent of nilpotence class c if , c = max{r Fr7r {1}} is finite. Thus rc±k7 = [7,rc-kk-17] j for §8bis3

k > 1.

If we view 7 as a group on which 7 acts by conjugation, then 1'2 7 = [7, 7] is the smallest normal subgroup for which the quotient has an induced trivial action. We generalize this idea to the action of a group 7 on a module M.

8b'.3. Nilpotent spaces and groups

345

Definition 8bis.17. The lower central series associated to the action of a group it on a module M is the series of submodules

•• • c rr,.m c

c • • • c r,2 m c m

where r„2m is the E7-submodule of M generated by elements of the form am — m for a G it and m G M, and Fir m = r2„.(P-„- i-m). We say that it acts nilpotently on M of nilpotence class c if c = max{r Fir m {c}} is finite. To a group or to a group action on a module, we associate various completion functors: = 11 m7/Fr7,

M = lim

m.

By the universal property of the inverse limit there are canonical homomorphisms j: it —> Fr and j: M M. Define Fit = ker(i:

it

Fit = coker(i:

FM = ker(i: M —>

—> it

—> Fr)

r'Do m = coker(i: M —> M).

By Lemma 3.10, r.„."m = nr r;m. Notice that, for a nilpotent group or nilpotent group action, i is an isomorphism and the functors r" and rioo vanish. Before turning to some examples, we introduce one more functor. A submodule N of a 7-module M is said to 7-perfect if r2,N = N. If we take the family Pm of all 7-perfect submodules of M, then we can define

rm = submodule generated by the union of the family PM . We leave it to the reader to show that rm is also 7-perfect; rm is the maximal 7-perfect submodule of M as it contains all 7-perfect submodules. By

construction, rm c r ir"m. A nice example of a nonnilpotent group action is given by our example 7rn (S2m x RIP271 ). Here the generator of 71 (S 2m x RP 2n) = 2217, acts on 2', by 1 —1. The lower central series for this action is seen to be • • • C 2r E, C 2 r-1 Z C • • •

c 27, CL

and so the action is nonnilpotent. The completion of L is the group of 2-adic integers, FZ 2z Z = {0} = FL. Notice that rico z is an uncountable group. If M is a module over the group 7, then we develop another expression for the terms in the lower central series for M. Let /7 denote the kernel of the augmentation E: 17r —> L, given by E(Eg ,„ ngg) = Eg,„ ng.

346

8bi'.

Nontrivial fundamental groups

Lemma 8bis.18. F M = I7r • M, that is, F M is the submodule of M generated by expressions of the form xm with x E hr and m E M.

PROOF: Since FM is the submodule of M generated by am — m with a E 7r and rn E M, we can write am — m = (a — 1)m E 1 -7 • M.

Thus

r2,M

c hr •

M. To obtain the reverse inclusion, take any element

a = EgEir ng in 7,7 7r and consider E geir ng g — E gE ir n 1. This is an element

in hr. For any element m in M we can now write gEir

ng (g — 1)) • m =

gEir

2 ng (gm — m) E FM.

It remains to show that every element of hr has the form Ege7t. ng (9 — 1). To see this, write a = Ege7t. ngg = n 1 1 + Ege7t., g1 ngg. When a E /7r, ng = —n i , and so we ean write a = ngg—E g„,g , n9 1. Thus /7r M c 177,2 M. Corollary 8bis•19• If 0 —f K M Q 0 is a short exact sequence of 7-modules and 7-module homomorphisms, then M is a nilpotent 7-module if and only if K and Q are.

We next introduce the topological category of interest. Definition es.20. A pointed connected CW-complex (X, xo) is said to be nilpotent if 71 (X, x0) is a nilpotent group that acts nilpotently on 7F,, (X, x0) for all n > 2.

The category of all nilpotent spaces together with continuous mappings . contains all simply-connected spaces, all H-spaces (Corollary 8 bis 3), and the Eilenberg-Mac Lane spaces K(7r, 1) for 7r nilpotent. As we will soon see, this category is also closed under certain constructions of importance in homotopy theory. There are spaces that are not nilpotent. Some general constructions are possible: (1) closed surfaces of genus greater than one; (2) the wedge product X V Y of any two non-simply connected spaces X and Y; (3) if X has fundamental group 7F and there is an epimorphism 7T G with G finite and G acts nontrivially on the rational homology of the covering space of X corresponding to the kernel of 7r > G, then X is not nilpotent. We begin our investigation of some of these constructions with a useful analogue of Corollary b8 is.19. —

8b'.3. Nilpotent spaces and groups

347

Proposition 8bis•21• If F E—> P B is a fibration with F connected and E nilpotent, then F is nilpotent.

PROOF: We can apply Lemma 8bis .18 inductively to show that FM = (1-7)' • M. Suppose that the nilpotence class of 7rn (E) as a module over it = 7r1 (E) is c. Then r„c+ 1 7rn (E) = {0 } , and so (t7r)c • 77„(E) = {0 } . Suppose that w E 7ri (F) and a E 7n (F) for n > 2. Then w • a = (i * w) • a in terms of the 7rj (E)-module structure on 7r„ (F). Suppose that w E (/7)c. Since i * is a in (E)-module homomorphism, i * (w • a) = (i * w) • (i * a) = 0. Thus w • a = 00 for some E 7 +1 (B). Suppose that n E 7ri (F). Then i o E 7ri (E) and a((i o - 1) • 13) = (i o - 1)(00) = (i* t7 — 1)(w

a) = (77 - 1)(w a).

However, 71(E) acts on 7rn+ (B) viap., and so (i * n —1) • 13 = (Nio —1). = — )3. Thus n • w • a = 0. This shows that (uni (F))c± 1 • 7r,.„ (F) = {0 } and so the nilpotence class of 7r(F) over 71 (F) is less than or equal to c ± 1. When n = 1 the argument for 7rj (F) acting on itself by conjugation is similar and left to the reader. 111 We next present one of the first results on nilpotent spaces, due to [Whitehead, 0 W541 in a paper in which the nilpotence class is related to the Lus temikSchnirelmann category (for the purposes of proving the Jacobi identity for the Whitehead product!). Theorem 8bis•22• Suppose (A, ao) is a finite, connected, pointed CW-complex and (X, x 0 ) is a connected, pointed CW-complex. Suppose f: (A, a0 ) —> (X, x0) is a fixed pointed map. In the compact-open topology, the space mapf ((A, ao), x0)) of pointed maps in the component of f is a nilpotent space. Furthermore, if X is nilpotent, then the space of unpointed maps mapf (A, X) is nilpotent.

PROOF: We proceed by induction. We assume that AO ) , the 0-skeleton of A, consists of a single point, 0,0. Then the theorem is seen to be true for mapf ((ao, ao), (X, x0)) and mapf (A (°) , X). 4 (n+ 1 ) that determines the Consider the cofibration V Sri —> A(n) i= i addition of cells to the next skeleton. Apply the functor map f (—, (X, x0)) to obtain a fibration (see §4.3): mapf ((A(n+ 1) , ao), (X, x0)) —> map f ((A (n ) , ao), (X, x0))

ma Pfl ((Vi_18n,*), (X, x0)).

By induction, we assume that mapf ((A(n), a o ), (X, x 0 )) is nilpotent. Proposition 8.21 finishes the proof. 111

348

8b''. Nontrivial fundamental groups A generalized Whitehead Theorem

When does a mapping f: X —> Y of connected spaces that induces an isomorphism of homology groups induce an isomorphism of homotopy groups here taken as a family of 71-modules? An answer to this question would generalize the Whitehead theorem (Theorem 4.5) to non-simply connected spaces. The functors introduced to study the action of a group on a module in fact give the key to the answer. The following theorem of [Dror711 helped to establish the category of nilpotent spaces as an appropriate category for homotopy theory. Theorem 8.23 (the generalized Whitehead theorem). Let f: X —> Y be a H. (Y) is an map of connected pointed spaces such that 11.(f): 11,,(X) isomorphism. Then 7r*(f) is also an isomorphism if, in addition, f satisfies the three conditions: (1) r ()°7* (f): ker(7„ (X) > 7r:(X)) > ker(7* (Y) —> 7 r TY- is an epimorphism. (2) rico 7r., (f) : coker (7r (X) —> 7r(X)) —> coker(74 (Y) 7r>,(Y)) is a monomorphism. (3) F7r* (f): Fr(X) r7r*(Y) is a monomorphism. —

—

) )

Theorem 8bis .23 fixes the role of the various limit functors. For example, if X and Y are aspherical spaces and f : X —> Y is a map for which r7r* (f) is an isomorphism, then f is a homotopy equivalence. When the fimctors r„", ri„ and F all vanish on 7r* (X), then the completion homomorphism 7r* (X) —> 7r* (X) is an isomorphism and the space X is called 7-comp 1ete. When a map between 7r-complete spaces is a homology isomorphism, it induces an isomorphism of homotopy groups. All nilpotent spaces are 7r-complete. Corollary 8 s.24. If X and Y are connected nilpotent spaces and f: X —> Y is a mapping that induces an isomorphism on integral homology, then f induces an isomorphism of hornotopy groups as graded modules over their fundamental groups.

To prove the theorem we sneak up on it inductively. Let Sn denote the collection of statements: ln . 71-1 (f) : 7r1 (X) 7r1 (Y) is an isomorphism of 7 1 -modules for 0 < j < n 1. 2,. H,(f): H(X) H(Y) is an isomorphism and H (f) is an epimorphism. 3n . r e° 7,1 (f): P (x) 7rn (X) —> 17 (y) 7,1 (Y) is an epimorphism. 4n . 1'00 7,(f) : r7rn (x) Poo 7rn (Y) is a monomoiphism. 5n . F7rn (f) : r7rn (x) r7rn (Y) is a monomorphism.

8'.3.

Nilpotent spaces and groups

349

The condition S1 holds by the assumptions of Theorem 8bis .23. We claim that S, implies that 7r,„ ( f) is an isomorphism and hence that Sn+1 holds. Proving this claim gives us the theorem. We first prove— that, among Sn , the statements 1„ and 2„ imply that 7rn (f): 7„,(X) 7r, (Y) is an isomorphism. This follows from two remarkable lemmas due to [Dror71], the first of which extends Theorem 8bis .16. Lemma es.25• Suppose 0: M M' is a homomorphism of modules over a group 7r. If Ho (): Ho (7r, M) Ho (7, M') is an isomorphism and Hi (): H1 (7r, M) H1 (7r, M ') is an epirnorphism, then 0 induces an isomorphism 71411'„ M Mi /TT, M' for all r > land hence, an isomorphism 0: _11? of completions.

PROOF: To the short exact sequence of 7-modules

.7t4- mx,im o,

O

there is a long exact sequence of homology groups, ending with Hi (7, M)

Ho (7, M) H0 (7,

=

Hi (7, M/11M)

—>

Ho (7, 11M) —> Ho (7, .111) —> Ho (7r, myr, 2 m).

m/rm, and, since MAIM has a trivial 7-action,

m/r„.2 m)

Ho (7) 0 m/r1m z m/r1m

Ho (7, M).

This shows that the last homomorphism in the exact sequence is the isomorphism of coinvariants Mir (M,)„. The interesting part of the long exact sequence (ir, m/r,2 m) Ho ( r, r,2 m) O. becomes Hi (7, M) —> Suppose Hi (0): Hi (7, M) —> Hi (7, M') is an epimorphism and that 0 induces an isomorphism 0: m/r„.2 m Ay/Gm'. Then we have the diagram

H1 (7, .A4-)

H1 (7, m/rm)

r„2 m/r7,2 (Gm)

o

Hi (7r, 1-141 )

Hi (7r, .714 -7,1m, )

rir2 myrir2 (orm,)

O.

The Five-lemma implies an isomorphism l'M/177,3 M 11,M71-1 Mi. By the same quotient argument as in the proof of Theorem 8bis .16, we see that 0 induces an isomorphism M/I1M ilr/r71.1141 . The lemma follows by applying the same argument inductively.

350

8b''.

Nontrivial fundamental groups

Corollary 8bis•26• If M —> M' is a 7-module homomorphism, M and M' are nilpotent, and 0 induces an isomorphism H0 (0) and an epimorphism H1 (0), then M and M' are isomorphic.

The next lemma provides another step in proving the generalized Whitehead Theorem. P„X —> Lemma 8bis.27• Suppose X is a connected space and K (77i (X),n) Pn _ i X is the nth fibration in the Postnikov tower for X. Then there is an exact sequence, functorial in X, given by 11,1+2(PnX)

/11( 7r1 (X), 7rn (X)) —› Hn+i (Pri X) Hn+2(Pn-1X) H„±i(P„_iX) —> (7r„ (X)), H„(X) —> H„(P„_i X) —> 0

PROOF: The Leray-Serre spectral sequence for this fibration has E2 -term given by Ep2, Hp (Pn _ i X;H q (K (7,-„(X), n)), where the action of it = 7ri (X) on 7r„ (X) determines the local coefficients. Since H,, +1 (K (Tr, (X), n)) = {O} (a consequence of Lemma 6.2) and K(7r„ (X), n) is (n — 1)-connected, we get a lacunary E2 -term in bidegrees (*, i) for i < n +1—there are only two nonzero stripes in bidegrees (*, 0) and (*, n). As in the derivation of the Gysin sequence (Example 1.D) we get short exact sequences cr+1 0 —> g°±1,0 —> En2 +1,0

O

43,7,

Ezn o

En2+2,0

Ei7n

0 —> Ej7„ —> Ha+i(P,,X) —> E

1 , 0 —> 0

g°±2,0

Splicing these together we get Hn+2(Pn X) —> H ri+2(Pn _1X) —> Hn±i(P,,X) —> Hn±i (Pri_i X) > Ho (Pn _i X ; 7-17,(K (7 7,(X), n))) —> Hn (Pn X) —> Hn (Pn _ i X) —# 0

However, from Proposition 8bis .4 and Theorem 8bis .10 we know that f (7n(X))1 "

Hi (7, 77, (X)),

= 13,

i = 1.

By the definition of a Postnikov tower, we have that H„(P„X) =H,, (X). Furthermore, H„(X) > H„,(1=)„_1 X) because H„(Pri_iX) = Er2i 0 = E 0 . The lemma follows after we make these substitutions in the exact séq uence.

8b'.3. Nilpotent spaces and groups

351

We now complete the proof of Theorem 8bis .23. Suppose that our map f : X —> Y satisfies the conditions S. Then f induces a map of Postnikov towers and by naturality of the short exact sequence of Lemma 8his .27 we get a morphism of exact sequences (X)

1-1-r, (Pr, _ X)

0

Hn+ i(Pn Y) Hn ±i (Pn _iY) — (7rn (Y)),_+ H(Y) H n (Pn_iY)

0.

fin+i(Pri X) Hrt±l(Pn —1X) —> ( 71-n (X ))ir

I=

n

fl

The leftmost horizontal map is seen to be an epimorphism by considering the next stage of the Postnikov tower where we have Hn+i(X) >> Hn±i (PriX), and similarly for Y. Since H +1 (f) is an epimorphism by 27„ we get the first vertical epimorphism. By the Five-lemma, (7„, (X)), > (7,(Y)), is an isomorphism. Next consider the other end of the exact sequence: —

Hn+2(Pn _ 1 X) H 1 (7r1 (X),7,X) H n+t(P,X)

—

epi rin+2(Pn-1Y)

-111(7101, 7rnY)

rin+1(PnY) rin+1(Pn—1Y) —> 7rn(rir •

The Five-lemma implies that H1 (71 (X), 7rn (X)) — H1 (7ri (Y), 7V (Y)) is an epimorphism. By Lemma 8bis .25 we have that 7„,(f) induces an isomorphism between 7rn (X)/Fr,7r„ (X) and 7r,„ (Y)/r;7r, (Y) for all r and hence induces an isomorphism 7r— n (f). Finally we use the remaining conditions of S. There are exact sequences of functors given by 0 —> r71-71 —> 7rn —> 7-E;; —>F o 7Vn—> 0

o

r7r„

7rn —> 7rn /r7rn —> 0.

The Five-lemma and conditions 3n , 4n , and n for 7rn (f) imply that 7rn (f) is an epimorphism. To prove that 7r, (f) is a monomorphism, we use 5„, that is, r7rn(f) is a monomorphism. We only need to show that 7rn X/F7rn (X) —> 7rn (Y)/r7r n (Y) is a monomorphism. The lower central series has the property that Fir m c r'm is always strictly decreasing until it becomes stable. This is because rçr m = 177,2 (r-7r- 1m). We also know that FM c rp4 for all r. In fact, this inclusion extends to r, any transfinite ordinal, as follows: If a = + 1 are ordinals, then let FM = F„2 (F!?,M); if a is a limit ordinal, let FM = n 0
8/"'. Nontrivial fundamental groups

352

shown already that 7r,, (X)/1;7r, (X) —> 7r,„ (Y)/1;7r,„ (Y) is an isomorphism for finite r. Introducing the limit ordinals, we get an isomorphism for r = 0.) and the argument of Lemma 81"1 S • 25 works for the higher ordinals. Thus, 7,2 (f) induces an isomorphism 7rn (X)/F7rn (X) —> 7rn (Y)/r7r,„ (Y) and so, by the Five-lemma, 7.„(f) is a monomorphism. A characterization of nilpotent spaces

In Chapter 4 (Theorem 4.35) we constructed the Postnikov tower of a space and stated that, for simply-connected spaces, the fibrations in the tower could be taken to be principal, that is, each p: Pn X Pn _iX is a pullback of the path-loop fibration over the Eilenberg-Mac Lane space K (7, (X), n + 1) via a k-invariant, : Pn _i X —> K (rrn (X), n 1): K(7, (X), n)

K (7,2 (X), n)

11

Pn X

PK (7rn (X), n + 1)

1

1

P,2 _ 2 X

K(7,, (X), n 1).

P

We next give a proof of this property of simply-connected spaces and generalize it to nilpotent spaces. Lemma 8bis.28. Let Abe a finitely generated abelian group and let E and B be

-+

spaces offinite type. A fibration K(A,n) E is principal if and only if it is simple, that is, the action of 71(B) on K (A, n) is trivial.

PROOF: Let's assume that p: E —> B is principal and it is pulled back over a classifying map : B K (A, n + 1). The relevant part of the long exact sequence of homotopy groups may be written P4

0 7r + 2 (E) -

r+i (B)

=1

(K (A, n + 1))

7,„ (E)

72„ (B) 0

=1

7,„ (B)

O.

The action of 7r1 (B) on A can be identified in the second row with the action of the fundamental group of the total space of the fibration û on the base space K (A,n + 1). But this factors through the action of the fundamental group of n + 1), which is trivial. Hence, the fibration is simple.

8b'.3. Nilpotent spaces and groups

353

Suppose next that 71(B) acts trivially on A = 11-n (K (A, n)). Consider the cohomology Leray-Serre spectral sequence for the fibration with coefficients in the abelian group A. Then, E2° 'n ^= H" (K (A, n); A) contains the fundamental class i corresponding to the identity map on K (A, n). Since K (A, n) is (n — 1)connected, the first differential to arise on i is the transgression da+1, and this gives a class 4+ 1(0 = [0] E Hri+1 (B; A); we can form the pullback over 0: B —> K(A,n + 1). This produces a space E0 together with a mapping g: E E0. Checking the long exact sequence of homotopy groups, g induces an isomorphism on homotopy, and so, in the category of spaces of the homotopy type of CW-complexes of finite type, g is a homotopy equivalence, and p is a principal fibration. It follows immediately from the lemma that a simply-connected space X has a Postnikov tower of principal fibrations. For an arbitrary space X, let {P,X,p n , } denote its Postnikov tower. We say that p: Pn X Pn _ i X admits a principal refinement if there is a sequence of principal librations PnX = Pn,cX

Pn ,2X

Pn,c—iX

Pn ,1X = Pn _ 1 X

with pn = q2 o g3 o • • o qc . With this extension of the notion of a principal libration, we can now give a characterization of nilpotent spaces. Theorem 81*.29. A space X is nilpotent if and only if every stage of its Postnikov tower admits a principal refinement.

PROOF: Since each qj is a principal libration, we can write its classifying map as n + 1). We proceed by induction. By the : properties of a Postnikov tower, (Pro X) = 7r 0 (P0 _1X) = {0} and so ri (X) acts trivially (hence nilpotently) on 7r0 (Pn ,i)Ci. Suppose that 71 (X) acts nilpotently on 7rn (Pn ,1 _ i X) of nilpotency class < j — 1. View the kinvariant as a fibration (up to homotopy) and q1 : Pn ,i X —> Raj _ i X as the inclusion of the fibre. By Proposition 8bis .21, 71 (X) acts nilpotently on (P,,,i X) of class < j. By induction, 71 (X) acts nilpotently on 7„, (Pn , cX) 7r0 (P0 X) 7(X). Suppose that X is a nilpotent space and 7 = ri (B). series for 77, (X) as a 7-module has the form {0} c rc,7„ (X) c

(X) c

The lower central

c L7,2 7„ (X) c 7r 1 (X).

By construction each quotient r„70 (x)/r„+170 (x) is a trivial 7-module. Consider the fibration pn : Pn X Pn _ i X . The homology Leray-Serre spectral sequence (Lemma 8bis .27) for this fibration gives the exact sequence Hri+i(PriX)

Hn+i(Pn_iX)

cr+1

354

8b''.

Nontrivial fundamental groups

Consider the cohomology Leray Serre spectral sequence with coefficients in (71-„X), = 7r„ (X)/17,2 7„ (X) for which -

4'q

HP

; 7-0 (K (77,(X), n); (7 7,(X))„)).

There is a class j Hri (K (7r, (X), n); (7r„(X))„) that represents the quotient 7r(X) —> (7r(X))„. This class transgresses to a class [12] lying in cr+1

which represents H.,±1(Pn -1 X) —> ( 7rri(X ))7tX —> K ((71- (X)),, n ± 1). and for which we take a representative 12: Let q2: Pn ,2X X be the pullback of the path-loop fibration over 12 and let u2: PriX Pfl ,2X be a lifting of pn through Pri ,2X. Such a lifting exists because 12 0 pri *• We can modify u2 to be a libration and consider a portion of the homotopy exact sequences Iln+1 (Pn _ I X; (7r(X))„)

172,77, (X)

0

7r?

(X)

-= 7r,„ ( fi bre (u2) )

n (Pn X) U2,

0

(7r, (X))

> 7r N(Pn ,2 X)

12. 7r, (Pr, X) —>

.

From this diagram we see that the fibre of u2 is K(r„2 7r (X), n). If we repeat this construction with u2 replacing An , then we get a space Pn ,3X together Pn ,2X. Continuing in this way, if X with a principal fibration q3: is nilpotent, we eventually get to F 1 77, (X) = {0} and the process stops with ue = qc and pn refined by principal fibrations. The sequence of k-invariants that a tower of principal fibrations admits may be applied to many problems in classical homotopy theory. For example, the kinvariants are the data for classical obstruction arguments. Another application was introduced by [Sullivan711 in his work on the Adams conjecture. [Serre531 showed, in his development of classes of abelian groups, that homotopy theory can become simpler when viewed one prime at a time. Making this notion topological rather than algebraic is the goal of localization at a prime. To localize a space X at a prime p, first consider the ring of integers localized at the prime p, denoted Ep, and given by the subring of Q2 of fractions a/ b with b relatively prime to p. The functor on abelian groups, A A 0 Ep , is called localization at the prime p; it eliminates all torsion prime to p and so leaves only the p-primary data. This functor can be extended to spaces by modifying the refinement of the Postnikov tower by composing the classifying maps 0,,i with the mapping induced by the localization, K(A n n) K (A n .i 0 2Zp , n),

8b's.3.

Nilpotent spaces and groups

355

and then pulling back carefully. The resulting space X, has homotopy groups H,(X)0Z p . 7r,„(Xp ) 7r (X)0; and integral homology groups II (X„) Later in the chapter, we will present an alternate construction of the localization of a space, due to [Bousfield-Kan72] and carried out simplicially. Convergence of the Eilenberg-Moore spectral sequence

®

Theorem 8bis .23, the generalized Whitehead theorem, illustrates how the nilpotence condition can control the effect of the fundamental group. The relations between the homotopy groups of a space and their nilpotent completions provide the data for measuring the departure from the simply-connected case of the Whitehead theorem. Another naive situation in which simple connectivity plays a role is the convergence of the Eilenberg-Moore spectral sequence. The goal of this section is to prove the following result of [Dwyer74] that shows how the nilpotence of a certain action of the fundamental group is decisive in generalizing the naive convergence criterion. Theorem 8 1*.30. Suppose F E—> r B is afibration with all spaces connected, and A is an abelian group. Then the Eilenberg-Moore spectral sequence for the fibre of p converges strongly to H(F; A) if and only if 71(B) acts nilpotently on Hi (F; A) for all i > O.

Following [Rector71] (§8.3) we associate to the pullback data X — B E the cosimplicial space (the geometric cobar construction) G• (X, B, E) where Gn (X, B , E) = X>< B" x E for n > 0 and with coface and codegeneracy maps given by (x,f(x),N,... ,bri,e) di (x, b1, • . • , bn, e) = { (x, b1,

• • • , N, bi, • • . , bn , e)

(x,bi, • • • ,bn,1)(e), e) 8i (\ x,

h .. • ,bn,e) _1,

— (x, b1, • • • , bi+ 1 , • • • , bn, e),

i = 0, 1 < i < n,

i = n + 1. 0 < i < n — 1.

In this discussion we take all spaces involved to be simplicial sets. Thus G• (X, B, E) is a cosimplicial simplicial set We explore the combinatorial structure of such an object in what follows. Let A denote an abelian group and X, a simplicial set (§4.2). Then we define the simplicial abelian group A ® X by (A ® X),„ = 019) xex A, for n > 0, with face and degeneracy maps induced by the maps on the generators and extended to be A-linear. It follows that 77. (A 0 X) IL. (X; A) and problems concerning homology become open to homotopy methods. In homological algebra, the basic datum of a resolution of a module M is the augmentation F. -> 6 M O. We can view a cosimplicial space -17* as

356

8/"'. Nontrivial fundamental groups

a kind of resolution (for example, when constructed from a triple; R3ousfieldKan72, LOD. We consider all possible augmentations of Y', that is, maps Y° satisfying d° o e = d' 0 E. The maximal augmentation associated e Z to Y' is the subspace aY' of Y° that gives the equalizer (as simplicial sets) of the coface mappings d°, : Y° —) Yl. In detail, the space aY• is given by aY' = {y E Y° d° (y) = (y)}. The maximal augmentation has the following characterization in the category of cosimplicial spaces.

Lemma es•31• The maximal augmentation aY* of a cosimplicial space 1' is the simplicial set CoSimp(*, Y') of cosimplicial maps from the constant cosimplicial space *.

We leave the proof of the lemma to the reader. The Hom-set of cosimplicial maps between X' and Y', CoSimp(X• , Y*), has the structure of a simplicial set with n-simplices given by the cosimplicial maps A [n] x X' —> Y'. Here A [n] , denotes the standard simplicial n-simplex, whose s-simplices are given by A[n] s = {(xo, xi, , xs ) 0 < xo < xi < • • • < x s < n}. The face and degeneracy maps on CoSimp(X*, Y') are induced by the standard maps. The inclusions Ei : A[n] —> A [fl + 1] are given by Ei (@0, Xi, ... x,$ ) = X (), X i , , X.0, where Xi = xi, if j <j, and Xi = xi 1, if j > i. The Ei X

face mapping is given by di : A [n] x X +1]x X' Y'. The degeneracy maps are defined by the combinatorial collapse onto the i th face, namely A[n] —> A [n — 1], given by ni ((xo, xi, ... • x)) = X(), -X-1, • • • ,

ni xi where Xi = xl, if / < j, and Xi = x 1 - 1, if / > j. Thus s3 : A [n] x X' —> A [n — 1] x X' —> Y'. A desirable property of resolutions is homotopy invariance. For cosimplicial spaces, we want a similar property—if f: Y' —> Z' is a morphism of cosimplicial spaces that satisfies the condition that f : Yn —> Zn is a homotopy equivalence of simplicial sets for all n, then f ought to induce a homotopy equivalence of maximal augmentations. However, this is too much to ask for. The fix for this desideratum is to replace the construction of the maximal augmentation with one that is more robust homotopically.

Definition es.32. Given a cosimplicial space Y', let Tot (Y') denote the simplicial set CoSimp (A' , Y') where A' denotes the cosimplicial space with A [n] at level n and coface and codegene racy mappings induced by the canonical face inclusions, Ei, and projections, nj respectively. ,

This functor was introduced by R3ousfield-Kan721 and forms the basis for their study of localization and completion. Tot (Y') can be built up canonically from a tower of fibrations. Let A,' () denote the s-skeleton of the cosimplicial space A', that is, at level n, one takes the s-skeleton of A [n]. Define

8b's.3.

Nilpotent spaces and groups

357

A• (.9± 1) inTots (Y-') = CoSimp(6,* (s) , Y'). The cofibrations L• duce fibrations Tot s+] (Y') —> Tot, (Y'), whose inverse limit is Tot„ (Y') = Tot (Y'). Notice that Toto (Y') = Y ° , and if 6: Z —> Y • is any augmentation, then e induces a mapping Z —> Tot s (Y-' ), for all s < DC. A tower of fibrations gives rise to an exact couple based on the long exact sequences of homotopy groups. The E'-term is determined by the homotopy groups of the fibres of Tot, —> Tot,_ ] . A typical fibre takes the form ((NY )8) where (NY) 8 may be written as Ys Clker s° n- Clker ss -1 when the simplicial sets at each level of Y are fibrant (that is, Yn —> * is a fibration for all n). It follows from the grading for the exact couple that this is a second quadrant spectral sequence. There are general conditions for its strong convergence to 78 (Tot(Y * )) (see [Bousfield-Kan72, IX, §51). We will obtain the Eilenberg-Moore spectral sequence in this manner by taking the homotopy spectral sequence associated to the tower of fibrations {Tot s (A 0 G* (X, B, E))}. In the category of cosimplicial spaces we find the usual notions of homotopy theory such as fibrations, cofibrations, and homotopy equivalences. The case of interest is the following diagram depicting a fibration of cosimplicial spaces along with an augmenting fibration of spaces: F

G*(*,B,E)

E

G* (B , B , E)

pl

iq B.

Here B denotes the constant cosimplicial space with B at all levels and the identity map for all coface and codegeneracy maps. The maps for G* (B, B, E) are given by id :B—>B<—E:p. The mapping q is given by first projection off the product B x Bxn x E. Thus, at each level, we have a trivial fibration and so 71 (B) acts trivially on each fibre Gn(*, B, E). We next show that the action of 7r1 (B) on H, (F; A) is compatible via the augmentation with this trivial action. Proposition 8bis•33• Ike augmentation map

(*, B, E) inE: F —> * x E = duces a -n-i (B)-equivariant homomorphism e * : H* (F; A) —> H* (E; A), where 71(B) acts trivially on II* (E; A)

PROOF: We argue with spaces and lifting functions as in §4.3. The simplicial versions of these structures can be found in [May67]. The pullback spaces for the fibrations p and q are given by Qp = (A, e) A E WB,e E E, A(0) = p(e)} , and St, = {(A, b, e) A E WB,(b, e) E B x E, A(0) = 131. The augmentation

8b''. Nontrivial fundamental groups

358

maps induce a mapping between these pullbacks 9, p (A, e) (A, p(e), e). This gives rise to the diagram S 2B x F

QB x (* x E)

A

St,

St, A ,

S2,, given explicitly by evi

WE

W (B x E)

evi

(* x E).

As described in §4.3, the lifting function for the trivial fibration is given by A' (A, b, e) = (A, ce ), where ce denotes the constant path at e. Since the action of (B) is induced by these composites, compatibility of the actions is equivalent to the homotopy commutativity of this diagram. Let H : S-2/3 x F x (* x E) be given by H((o.), y), t) = (*, A(o.), y)(t)). Then H makes the leftmost square commute up to homotopy and so proves the proposition. The fibration of cosimplicial spaces G• (*, B, E) G• (B , B, E) B provides control of the 71(B)-action in the tower of librations that give rise to the Eilenberg-Moore spectral sequence. Lemma 8.34. For all i > 1, 7r, (Tot s (A ® C (*, B, E))) is a nilpotent 71 (B)module. PROOF: We prove this by induction over s. When s = 0, we have the trivial B that describes the 0-level of the fibration of fibration E B >< E cosimplicial spaces. Thus 71(B) acts trivially on 71-i (Tat° (A G* (* , B, E))). By induction we consider the fibration Tot n (A 0 G* (* , B , E)) —> Tot n _i (A

G* (* , B , E)).

[Bousfield-Kan72, X, §61 give an explicit expression for the fibre of this fibration from which we deduce its structure as a 71 (B)-module. To wit, the fibre of Totn (Y*) Totn _i (Y. ), for any cosimplicial space Y*, is given by the function space Hom( (Sa, *), (NY' ,*)) where NY n = ker(Yn ) and mn—ly• is the (n —1) st matching space consisting of simplices in (yn-1) X n, written (x ° , . . . , Jin-1 ), that satisfy s ixj = 1 xi whenever 0 < j < j < Ain— Yx• is given by y ( so y, s n t y) . n — 1. The mapping s yn —

In the case of a cosimplicial simplicial abelian group, the homotopy groups of the fi bre may be written 7ri (fibre(Tot„ (Y*) —> Tot„_i (Y. ))) 7ri+n (NY n ) = 7ri+,(Yn

)

n

ker s ° n • • • n ker sa ' .

When Y• = A 0 G• (*, B, E), the homotopy groups of the fibre have a 71 (B)-action inherited from the inclusion into A Ø G (*, B, E). However,

8b'.3. Nilpotent spaces and groups

359

(A ØC ' (*, B, E)) is a trivial in (B)-module, and so then are the homotopy groups of the fibre of Tot(Y) Tot n _ (Y*). By induction, we assume that the groups 71-1 (Totn _i (A 0 G(*, B, E))) are nilpotent 7ri (B) -modules. The long exact sequence of homotopy groups for the fibration Tot ri (Y*) —> Tot,_ 1 (Y') and the triviality of the 71 (B)-action on the homotopy groups of the fibre complete the induction.

From the lemma we can deduce half of the proof of Theorem 8bis .30. Suppose that the spectral sequence converges strongly to II (F; A). Then there is a filtration of Hi (F; A) for each i with E_p isomorphic to the associated graded group to this filtration. Strong convergence implies that the direct limit of the sequence •••

> 7ri (A0Tot s (*DC' (*, B, E)))

7ri (Tot s _ 1 (A0G*(*, B, E)))

—

—

>•••

vanishes and so there is an injection -

(Totp_ r. (A

R. = nr

G*) —> Tot, (A 0 G*)) C (Totp (A 0 C. (*, B , E))).

It follows that each E * is a nilpotent in (B)-module. Strong convergence also finite in number. Arguing implies that the nonzero 71 (B)-modules E _ inductively using Corollary 8bis .19 we have proved that Hi (F; A) is a nilpotent in (B)-module. To prove the other half of Theorem 8bis .30 we use the towers of fibrations that arise from the application of the functor Tots to the cosimplicial fibration G* (* , B , E) —> G* (B , B , E) —> B. The augmentation from the fibration p may be depicted in the diagram: F

Toto G* (* , B , E)

Tot i G* (*, B, E)

•• •

Tot G. (*, B, E)

E

Tot o G* (B , B , E)

Tot i G* (B , B , E)

•• •

Tot G* (B , B, E)

B

Lemma 8bis•35•

=

Tot(G" (B, B, E))

B

B.

E.

The projection off the last coordinate Gn (B, B, E) —> E provides an inverse to the augmentation E —> G* (B , B, E). PROOF:

It follows that we can compare the augmentation fibration with the limit The nilpotency condition plays a role in the following proposition that is a form of the Zeeman comparison theorem. The proposition was known in the early 1970's—it is stated explicitly by 11-1ilton-Roitberg761. fibration.

360

8/"'. Nontrivial fundamental groups

Proposition 8bis.36. Suppose F

E B and F' —> B are fibrations with B connected, and f: E E' is a map over B inducing an isomorphism on homology. If 71(B) acts nilpotently on Hi(F) and on Hi(F'), for all i, then F F' induces an isomorphism on homology.

PROOF: We proceed by induction on the degree i of Hi (f). In the case i = 0, Ho ( f) = Ho (fl) because B is connected. Suppose Hi ( f 1) is an isomorphism for 0 < i < n 1. This implies that the E2 -terms of the associated Leray-Serre spectral sequences are isomorphic in bidegrees (s, i) for i < n 1. We consider the morphism of spectral sequences in bidegrees (0, n) and (1, n), where we have Ho(B,R n (F)) Ho (B,'Hn (F')) E' 02 ,n . By Proposition 8bis .4, —

—

110(B :1-((F))

H0(7, Hn (F)) = (Hn (F))„,

= Hi (7, Hn (F)). On the where it = ri (B). By Theorem 8bis .11, E?, ^vertical edge of the spectral sequence the map of spectral sequences gives En+1 —Ec° On 0,n —

-1 OV = n

pc' • 0,n

Since the E2 -terms are isomorphic in bidegrees i) for i < n 1, the differentials arising to make the successive epimotphisms along the vertical edges are the same in each spectral sequence and so we conclude that Ho (7, Hn (F)) is isomorphic to Ho (7, Hn (F')) via Hri ( f 1). Similarly, we find that H1 (7r, Hn (F)) maps onto Hi (7r, Hn (P)) via _fin (f ) . Theorem 8 131s .16 implies that Hn (F) is isomorphic to Hn (F'), and the inductive step follows. —

The second half of the proof of the Theorem O s .30 follows because the homotopy spectral sequence for the tower of fibration {Tot s (A ® G• (s, B, E))} converges to 7F* (Tat (A. 0 G. (*, B, E))). Proposition 8bis .36 implies that 7F* (Tot (A 0 G• B, E))) H * (F; A). Theorem 8bis .30 has been extended to connective generalized homology theories (Mousfield871), nonconnected bases B (Mror-Farjoun—Smith, J901, a useful case when dealing with function spaces) and to pullback fibre squares Y for which the set 70(X) X 0 (B) 70(Y) is finite and, with data X —> B for all y E Y, 71(B ,p(y)) acts nilpotently on H* ((Y) y ) where (Y) 0 denotes the component of Y containing y (IIShipley961). The development of convergence criteria for the Eilenberg-Moore spectral sequence is, in fact, a spinoff of the investigation of the general convergence properties of the Boustield-Kan spectral sequence.

es.3.

Nilpotent spaces and groups

361

Theorem es.37. Given a fibrant, pointed, cosimplicial space Y, there is a spectral sequence associated to the tower offibrations {Tot (Y) Tot,„ (Y*)} with E t (Y)\

fg\

n ker

CI

•••

n ker ss -1 ,

t

> s >0

and converging under favorable conditions to 7F* (Tot (Y* )).

General results indicating favorable conditions were obtained by [Bousfield87], [Shipley96], and [000dwillie98]. The fundamental example introduced by [Bousfield-Kan72] is the cosimplicial space associated to the completion of a space with respect to a ring R. The R-completion of a pointed space (X, xo) is obtained by applying the totalization functor, Tot, to the cosimplicial space RX obtained from the triple {R, 0,1} as follows: If (X, xo) is a pointed simplicial set, then define the simplicial R-module RX by (RX),, = R ® X I R X. The natural [1 0 x], and the natural transformation 0x : X —> RX is defined by x transformation '0x : R 2X —> R X is given by [r 0 [s 0 x]] [r s x]. The R-completion of X is defined by R oc X = Tot (R*X) =

11* X),

where Rk X _ R(Rk-1,0) and R°X = RX . The cosimplicial structure is based on the natural transformations 0 and L', with the coface and codegeneracy maps given by R k X —> R k-"- X, 83

Rk

x

= =

(

OR k - ) 1

OPRk-, x )•

It follows from the properties of Tot that R, X is the inverse limit of a tower of fibrations R s X —> R s _ i X where R X = Tot s (R•X). This tower of fibrations is augmented by a family of mappings fs : X —> R s X and it leads to the spectral sequence of [Bousfield-Kan721. When R is a subring of Q, then, one can prove that, for some set P of primes, R = Zp = {a lb E (12 p b, for all p P } . The R-completion of a nilpotent space (X, x0 ) coincides in this case with its Ep -localization ([Bousfield-Kan72, V, §41). Thus (co)simplicial techniques generalize the localization construction via Postnikov towers of [Sullivan711 to general rings. The basic algebraic condition on the ring R that guarantees good completion properties is that R be solid, that is, the multiplication on R induces an isomorphism R R —> R. ; R) —> fl * (R oo X ; R) is an isomorphism, then we When f,: say that X is R-good. For R-good spaces the R-completion, f, : X —>

362

8/"'. Nontrivial fundamental groups

R„X, enjoys certain universal properties. For example, a mapping f: X —> Y induces an isomorphism 118 (f): H„(X; R) H„(Y ; R) if and only if R oo f : R oo X —> R„Y is a homotopy equivalence ([Bousfield-Kan72, 1.5.5]). However, there are spaces that are not R-good—for example, an infinite wedge of circles is not Z -good. Nilpotent spaces are Z -good. With this language we can describe the solution to the natural question—what is the target of the Eilenberg-Moore spectral sequence in general? [Dwyer75] found the answer for the Eilenberg-Moore spectral sequence associated to the fibre of a fibration p: E —> B: The spectral sequence converges to the homology of the nilpotent completion of the fibration, that is, to H8 (F; R), where F is the fibre of the fibration R oop: RE —> R„B.

Completion and localization constructions have become fundamental in homotopy theory and a complete exposition of these ideas would take us too

far afield. Nice expositions of this circle of ideas may be found in [Sullivan71], [Mimura-Nishida-Toda71], [Hilton75], [Hilton-Mislin-Roitberg75], and [Arkowitz76]. The most complete exposition of these ideas is the work of [B ousfield-Kan721 A consequence of the cosimplicial construction of the R-completion is a result of [Dror73] that shows the extent to which nilpotent spaces approximate general homotopy types. To state precisely what sort of approximation we mean, we compare a connected space X with the associated tower of fibrations {R,X}. By the definition of Tot s , we have the augmentation mappings f,: X —> R,X for all s > 0 and these mappings are compatible with the sequence of fibrations R 8+1X —> R,X. Thus the mappings {fs } determine a mapping of towers of spaces {X} —> {R,X}. A tower of groups {G, } is a sequence of homomorphisms G9+1 —> G9 for s > O. A homomorphism of towers of groups, C: {G 8 } —> {Hs }, is a sequence of group homomorphisms C8 : G, —> H,, compatible with the tower mappings. The natural maps f9 : X —> R s X determine, for each i > 0, a homomorphism of towers of groups f {H,(X ; R)} —> {H i (R, X ; R)}.

Definition 8bi8.38. A homomorphism of towers of groups, {G,} —> {H 8 } , is a pro-isomorphism if, for any group A, C induces an isomorphism : lim Hom Grp (Hs , A) —> limHom Grp (G 9 , A).

We leave it as an exercise to show that C: {C 8 } —> {H 8 } is a proisomorphism if and only if, for each t > 0, there is a value t' > t and a homomorphism Ut: Ht , —> Gt such that the following diagram commutes: Ht,

363

Exercises

Here P t denotes the composition t " • Gt+ Gt and likewise for /JP t . We also leave it to the reader to show that a pro-isomorphism induces an isomorphism of limits: Bill Gs lim Hs and : lim 1 G, lim 1 Hs . <_s We say that a pointed space (X, xo) is R-nilpotent if X is nilpotent and, for each n > 1, there is a central series of ri (X, x0)-modules 7rri (X, xo) =

M2

• • • Mc _i

Mc

Mc+i =

{0},

for which each subquotient .714-3 /M3+1 is a trivial 71 (X, x0)-module and an R-module. A space is nilpotent when it is Z -nilpotent. Proposition 81*.39. For an arbitrary connected, pointed space (X, x0), the spaces R s X = Tot s (R* X) are R-nilpotent for all s > O. Furthermore, the natural maps f: x 118 X induce, for all i > 1, a pro-isomorphism of towers of homology groups f s : {H,(X; R)} —> {H,(R s X; R)}. SKETCH OF A PROOF: The space RX is R-nilpotent since it is an H-space and an R-module. According to [Bousfield-Kan72, 1[1.5.5], if p: E B is a principal fibration with connected fi bre F and any two of E, B, and F are R-nilpotent, then so is the third. Their Proposition 112.5 asserts that R s X —> R s _ i X is a principal fibration with fibre a connected simplicial R-module. Thus the spaces R s X are R-nilpotent for all s > O. To establish that we have a homology pro-isomorphism, we observe that k (X ; R) k (RX X0) and so we can compare the tower of homotopy groups {7rk(RX, xo)l with {7rk (RR,X, x0)}. When comparing the spaces RX and RR s X , we have a triple structure available and hence mappings

0: RX

RR s X : IP

with 1/)(5 = id.

[Dror731 interpolates a condition that implies that a pro-isomorphism on homotopy is induced by {RX} {RR s X}, namely, that the map of towers {11X} {R s 11,,X} induce a pro-isomorphism. He then uses the convergence of the homotopy spectral sequence associated to the tower {R s X} to {Hk (R s X: R)} for k > 1. CI obtain the pro-isomorphism {Hk (X; R)}

It follows from the proposition that every connected, pointed homotopy type may represented by a tower of R-nilpotent spaces, up to homology equivalence. This approximation is analogous to the Stone-Weierstrass theorem: Every homotopy type (continuous function) may be represented by a tower of R-nilpotent spaces (a sequence of polynomials) such that II (X; R) ils (R s X; R) (limits agree). To study whether a space is R-good, we can focus on the relation between lim II (R s X; R) and I1 (R„ X; R). 4-8

364

8b, Nontrivial fundamental groups

Exercises p2n ). 8bis .1. Show that 7ri (IlIP 2n ) acts nonnilpotently on 72( n 8bis .2. Show that the action UnE ' F : 7r1 (Ee) x 77.„(F, e) e) is welldefined and that : 7rn e) —> 7rn (E, e) is a 7r' (E, e)-module homomorphism. 8bis .3. Suppose that M is a module over a group 7r. Show that the coinvariants is the largest quotient of M on which 7r acts trivially. Show directly that the functor M 1 > Mir is right exact. —

8bis .4. Let 7T denote the cyclic group of order m, with generator t E 7r. Show that

the complex 6

Wo —> —> 0

W2

is a resolution of over E7r, where Wk the free Dr-module on a single generator wk and boundary homomorphisms T: TV2n+1 > W27,, given by T( w2+1) = tW2ri W2n and N: W2 —> W2_ given by N( w2) = W2n-1 tW2n —1 ± —

•••±

tm —1 W2n

8biS.5•

Suppose that 7r is a finitely generated group. Show that Hi (rr, M) is finitely generated whenever X/ is finitely generated over E7r and j > O. 8bis .6. Prove directly that Hi (7r) `=" 7r/ [7r, 7r]. 8bis .7. Prove Theorem 8bis .11. 8bis .8. Suppose that 1 > H > 7 > Q > 1 is an extension of groups. Complete the proof of Theorem 81jis .14 by showing that the Q-coinvariants of the conjugation action on III[H,H] are given by H1[7r,H]. —

8 bis • 9.

—

—

—

Suppose that 1 >R >F > 7r 1 is a presentation of the fundamental group iv = 7ri (X) of a space X, where F and R are free groups. Prove the classic result of Hopf that H2 (X)1h, * (7, (X)) Rn [F, F]/ [F,R] where h* : 72 (X) > H2 (X) denotes the Hurewicz homomorphism. —

—

—

-2

8bis•0. Suppose that 1

—

> H > 7r > Q > 1 is a central extension, that is, maps to a subgroup of the center of 7r. Show that there is an exact sequence: —

—

H2 (70 —> 112

—

—>

H

—

—

> (7)

—

>

H

(Q) 0.

8bis .11 Prove Corollary bsis 19_ 8bis .12. Suppose that F B is a fibration of connected spaces. Suppose that E is nilpotent. Show that 7r1 (F) acts nilpotently on itself by conjugation. 8bis .13. Suppose 7r acts nilpotently on M and Ho (7r, M) = {0 } . Conclude that M = {0 } .

365

Exercises

8bis .14. The functor r associates to a 7-module the submodule LA/ generated by the union of the family of all perfect submodules of M, that is, submodules N with N = FN. Show that I'm is also perfect and that it is the maximal 7-perfect submodule of M. Show that I'm c rn,M for all n. 8bis •15. Show that rn,A4- = ri,±1m implies that rn,M = rir ri±km- for all k > O. Thus the lower central series is a sequence of proper inclusions until it stablilizes. 8bis •16• Show that all nilpotent spaces are 7r-complete. Show that the maximal augmentation of a cosimplicial space Y• is given by al" = CoSimp(*, Y').

8bis •18• If R c Q is a subring of Q, then show that there is a set of primes P (possibly empty) for which

R = Zap.

8bis •19• Show that a homomorphism of towers of groups, : {C 8 } — > {H s }, is a pro-isomorphism if and only if, for each t > 0, there is a value tl > t and a homomorphism Ut: Ht, Gt such that the following diagram commutes:

Gt , G

29 ,/

Gt

H.

Show further that a pro-isomorphism induces an isomorphism of limits:

: lim G

lim Hs and : lim l Gs lim l Hs.

9 The Adams Spectral Sequence In (various papers) it is shown that homological algebra can be applied in stable homotopy-theory." J. E Adams

One of the principal unsolved problems in modem mathematics is the determination of the homotopy groups, 7T* (X), of any nontrivial finite CWcomplex X. These groups play a key role in the solution of certain geometric problems and in the classification of CW-complexes up to homotopy. The computation of 7T* (X), however, remains difficult if not intractable. (For a discussion of the computability of 7r* (X) ® Q, see the paper of [Anick85]; for some interesting progress for X = Sri, see the work of [Wu, J]). A first approximation to 77* (X) is provided by the Freudenthal suspension theorem (Theorem 4.10). The limit groups, limk„ [Sn+k, SkX] — called the stable homotopy groups of X and they enjoy some regularity are and further structure. Knowledge of these groups may also be sufficient for the solution of geometric problems (§9.4). A classical example is the celebrated theorem of [Adams601 on the nonexistence of elements of Hopf invariant one (Theorem 9.38). As this theorem was part of the motivation for the construction of the Adams spectral sequence, we discuss some of the details. The question settled by Adams arose with W. R. Hamilton (see [Ebbinghaus90]): For which n, does Rn have a division algebra structure? That is, for which n is there a bilinear mapping, Rn x Rn —> Rn , so that 14/7i,, fl) = 0 implies that either it = 0 or it = 0. For n = 1, 2, 4 or 8, there are the real, complex, quaternionic, and Cayley multiplications, respectively, that were classically known. If one requires further that 11)-t(ES)11 = 11Y11119"11 (a normed algebra), then [Radon22] and [Hurwitz23] showed that these classical multiplications are the only examples. [Hopf31, 35 1 used the classical multiplications to construct mappings, /7: v: S 7 —> 54, and a: S 1-5 > S 8 , which are not homotopic to the constant map. These are the first examples of nontrivial elements in the homotopy groups of spheres (other than the degree maps in 9 77,( n )). Hopf's proof is geometric and proceeds, in the modern parlance, by introducing a homomorphism H: 72n-1(S") —> R, constructed by counting linking numbers of the inverse images of points in Sr' . Hopf showed that the —

367

linking numbers are a homotopy invariant and then applied the division algebra structure to deduce that the maps derived from the classical multiplications have H-invariant equal to one. One can also compute linking numbers with the cup-product in cohomology. [Steenrod49] studied the Hopf invariant, H 7 2Tt - 1( Sa ) —> R., Using functional cup products: Given y: S2' —> Sn, form the mapping cone, K = S U e2 . The Hopf invariant can be defined as follows: Let G Hn (K) and y2, G H2 (K) be generators for the free abelian group in each dimension. Then xn xn = ±11- ( -0 Y2n • When we reduce to mod 2 coefficients, we can make the transition to stable homotopy. If H('y) = 1, then xn xn = Y2n• By the unstable axioms for the action of the Steenrod algebra (Theorem 4.45), Sex ?, = y27,. We can suspend the map -y: S2n -1 Sri, and form the mapping cone. Then EK 5n+1 Liv, e2n+1 The suspension isomorphism determines H* (EK; R2) as a module over the Steenrod algebra. In particular, Sq n xn+i = Y2n+1 , where Xn+1 G Hn + 1 (EK; 72) and y2,±1 G H2n+1(EK ; 72) are generators. This implies immediately that E-y is not homotopic to the constant map. By iterating this procedure we see that if H(-y) = 1, then y determines a nontrivial element in _ 7ns_ i ( so _ the (n l) st stern of the stable homotopy groups of spheres. The existence of a division algebra structure on It', then, implies a nonzero element [y] exists in 7rns with the mapping cone exhibiting a nonzero Sqn operation in mod 2 cohomology. This reduction is already useful. According to the Adem relations (Theorem 4.45), Sqn factors into sums of products of lower degree Steenrod operations, unless n = 2 k , for some k. For example, the relation —

Sq 3 Sq 4 =

(4 — 1 — j )

Sq

7_3

5q3 ,

implies Sq 7 = Sq3 Sq 4 . It follows that Sq 7 cannot act nontrivially on the cohomology of the mapping cone, H* (Sn U f 0+ 7 ; 79). because Sq l through Sq 6 act trivially for dimensional reasons. We conclude, then, that R7 cannot carry a division algebra structure. In fact, Rn is a candidate for a division algebra structure only if n = 2k. To settle the division algebra problem, we are led to an analysis of Sq2k and the possible factorizations that might arise through secondary or higher order operations associated to the Steenrod algebra. [Adams 60 1 completed this analysis to prove that the classical examples of R, R2 , R4 , and R8 provide a complete list of real vector spaces with a division algebra structure. In the course of this work, [Adams60] also introduced his eponymous spectral sequence that has become one of the fundamental tools in the study of stable homotopy theory. The aim of this chapter is the construction and elaboration of this spectral sequence. The first section contains some motivation, a statement of the main theorem, and a discussion of secondary and higher order cohomology operations. Based

368

9.

The Adams Spectral Sequence

on the motivating discussion, we take a brief detour into homological algebra to introduce another important tool in the study of categories of modules over an algebra, the Ext functor. During this digression we introduce a change-of-rings spectral sequence. In §9.3 we construct the spectral sequence and derive its basic properties. In keeping with the spirit of previous chapters, we do not utilize the technology of spectra and the stable homotopy category ([Elmendorf-Kriz-Mandell-May97]). The reasons for this choice are as follows: The approach using spaces is contained in the original papers of [Adams58] and, though cumbersome, it can be understood by the novice. Also, there are now several careful and complete expositions of the spectrum approach, [Adams69], [Switzer75], [Ravene186] and [Kochman96], on which this author could not improve. The reader may safely skip to §9.4 if he or she accepts the main results and wishes to go quickly to the computations. The references, especially Adams's papers, may also substitute for this material. In §9.4 we explore some of the geometric consequences of the existence and explicit form of the spectral sequence. We focus on the role played by the Adams spectral sequence in computing cobordism rings (the work of [Thom54], [Milnor60], [Liulevicius62], and [Wa1160]). This section is written backwards—we take as basic the spectral sequence and search for applications. This emphasizes technique over the deeper geometric insight of [Thom54] and others. However, it gives a smooth transition into this set of remarkable results and offers a natural motivation for the study of spectra and stable objects. In some of the simpler, low-dimensional calculations are made and the geometric consequences explored. In particular, the first nonzero differential in the spectral sequence at the prime 2 settles one case of the division algebra problem. The low-dimensional stable homotopy groups of spheres at the primes 2 and 3 are also deduced. In the final section of the chapter, we consider further structure in the spectral sequence. A product structure allows one to define Massey products and these are seen to converge to the secondary composition products of [Toda62]. The structure of the Steenrod algebra as a Hopf algebra imposes homological conditions on the spectral sequence including a large region of the first quadrant where all of the input at E2 is trivial. There is also a periodicity operator that acts across part of the spectral sequence. The formidable task of determining the E2-term of the Adams spectral sequence is developed in §9.6. The tool of choice is the May spectral sequence, introduced by [May64] in his doctoral thesis. We describe this spectral sequence in §9.6. We close the chapter with some tables and a discussion of further applications. 9.1 Motivation: What cohomology sees

The computation of 71- >, X) or 7r? (X) is a special case of the more general problem of determining [Y, X] or {Y, X}k = Sn X] . Here we (

369

9.1. Motivation: What cohomology sees

assume that our mappings are basepoint preserving, X and Y are connected and of the type of CW-complexes and, finally, that Y is finite-dimensional. A naive "picture" of [Y, X] may be obtained by considering the image of the mod p cohomology functor, H*( p )

[Y, X]

Hom(H* (X ;I ), H* (Y ;

))

(Greenlees881 develops this idea for other cohomology theories). Because the mod p cohomology of a space carries a rich structure, this Horn may be taken to be a set of mappings in various categories. Of course, H* (f; Tp ) is a homomorphism of graded vector spaces. It is also a graded algebra mapping. This can be used, for example, to distinguish EP(2) from S2 V S4—the mod p cohomology of these spaces are isomorphic as graded vector spaces but not as graded algebras. Finally, H- (f; ) is a mapping of unstable modules over the Steenrod algebra—with this structure we distinguish ECP(2) and 53 V 55 • In the rest of this section, we fix a primep and let H- (X) denote H* (X ;7 2„). Let Ar denote the mod p Steenrod algebra and M and N graded left A p modules. Let HomtA, (M, N) denote the set of Ar -linear mappings of M to N that have degree —t, that is, for all q, f (Mg) c Ng' . The iterated suspension functor on graded modules over Ar is defined as follows: For k Z, s k : ModA i, —> ModA i, is given on objects by (sk M) Mn-k and on morThis generalizes phisms 0: M N by (s k O)n = the topological Ori-k. suspension isomorphism s: ill (X) fil+1 (S X). HomtA, (H* (X), H* (Y)) as folWe construct a mapping {Y, X } t lows: A mapping, f: Sn±t Y —> S nX, determines a moiphism H* (f) of modules over the Steenrod algebra. However, as modules over A r, i:/-* (SEX) s nfp(x) , sn-kt f f -17N) and fl*(f) determines a mapping (Sn±ty) in HomtA, (H* (X), H* (Y)). Furthermore, S f : Sn+ t+1 Y —> Sn+ 1 X determines the same mapping as in the diagram

fr-"-(sn+ix)

fi*(snx)

k s (sn+t y)

S

*+, (s n+t+, y) .

Thus H- ( ): {Y, X} t —> HomtA (H* (X), H* (Y)) is a well-defined mapping (the choice of pointed maps and connected spaces determines the mappings on H°). The image of this mapping may be taken as an approximation to {Y, A}t . It is, however, only a coarse approximation. A worst case is given by H- ( ): 7t—f 7, ) . The only classes in -R-is that are mapped nontrivially are generated by : S —> SE, the identity map in 74. The best case, however, suggests a course of action. Consider a free, left Ar-module

370

9.

The Adams Spectral Sequence

on one generator of degree n. This can be constructed as an extended module A p 0 snIFp where snIF2, is the graded Tp -vector space with one copy of IE , in degree n and {0} in every other degree. We introduce the notation"=' t for a homomorphism of graded modules that is an isomorphism in degrees less than t. A consequence of the Cartan-Serre theorem on H* (K(L I n);7 (Theorem 6.20) is the bounded isomorphism A p 0 sn7 p

* (K (7, I

n); 7 p).

This isomorphism leads to an isomorphism in the limit over the system of homomorphisms induced by the loop suspension mapping (Theorem 6.11) H1 (K (E pE, n t);Fp ), for n t < (K (lpz7,,n t 1);7 p) 1 < 2n 2t — 1: A p sn F

P

s —tfi* (KG,'

n + t); 7p)•

For Y a finite dimensional CW-complex and n < 2 dim Y, the fundamental correspondence [Y, K(/p, n)] Hn(Y; p) implies {Y, K (Z 73E, ti)}t = Hom t,t, Op

sn7p ,H* (Y; Ip )).

Thus our approximation is on the mark when we consider spaces that carry a free Ap -module structure, that is, Eilenberg-Mac Lane spaces. To increase the

accuracy of the approximation, we could include the information that measures how far a module M over Ap differs from being a free module. To do this, we introduce the functors Ext As (M, —), the derived functors of Hom 'A (M, —), to be discussed in §9.2. The reader should compare this discussion With §7.1 where the derived functors of the tensor product, M Or — are considered. The role of these derived functors is seen in the main theorem of this chapter, due to 1Adams581. For an abelian group G, (p)G denotes the quotient (p)G = G/{elements of finite order prime to p}.

It is elementary to show that the set {elements of finite order prime to pl forms a subgroup of G. Since {Y, X } t is an abelian group, (p) {Y, X }t makes sense. Theorem 9.1. For X and Y spaces of finite type, with Y a finite dimensional CW-complex, there is a spectral sequence, converging to (p) {Y, X } * , with E2terni given by (H* (X;T p ), H*(Y; Fp )),

and differentials dr of bidegree (r,r — 1).

371

9.1. Motivation: What cohomology sees

Before beginning our discussion of Ext and the construction of the spectral sequence, let us consider how one might show that a mapping f: Sn+t —> Sn is not homotopic to a constant map. Suppose t > 0 and we form the mapping u en+t+1. Then there is a cofibration sequence: cone, C(f) = Snf sn

include bottom cell

smash bottom cell

CU)

> sn_Ft+1

The exact sequence in cohomology, _>

(sn+t+

(c( f ))

(S n )

determines an extension of 7p by Tp over the algebra Ap and so a class in extA, (Fp ), here of degree t+ 1. This extA, ) is the classical Ext or Ext 1 group that figures in the Universal Coefficient theorem for cohomology ([Massey91, p. 314]). When one provides an abelian group structure on Ext, the correspondence between a representative of a stable mapping Sn+t —> Sn and the extension it determines gives a homomorphism (an e-invariant), e:

(S") —> extA, (F p ,

).

In the case of the classical Hopf maps, e(Hopf map) O. Whenever e([f]) 0, f-/-* (C(f)) is a nontrivial module over Ap. In general, the Steenrod operations on H* (C( f )) are trivial on two-cell complexes. It may be the case, however, that a secondary or higher order operation, coming from relations in Ap, is nontrivial on (C(f)). This also implies that f *. With this in mind, we next discuss higher order cohomology operations. Higher order cohomology operations

Suppose W is a space and : K (Z/ pE, n) —> K (E/ pZ, n t) represents an element in the Steenrod algebra. Suppose x E Hn (W; 4) is a cohomology class and that 0(x) = O. Under these conditions, a secondary operation can be defined. Let E denote the total space of the pullback of the path-loop fibration over K (ElpE, n + t) with respect to the mapping O. Let a: E K(Z/pZ, m) represent a class in Hm(E;Fp ). Since 0(x) = 0, x lifts (not necessarily uniquely) to a mapping X : W —> E. The set of all composites, ctoj-c E Hm (W ;F p ), varied over all liftings X, defines the secondary operation, (1),,,9(x) c Hm(W; 7p). a )

We make several observations about this construction:

K (17p2Z ,

372

9.

The Adams Spectral Sequence

1)If t < n, then [0] E Hn+t (K (2Z/pE, n); Ip ) is primitive and hence a loop map (0 = 00'; Corollary 8.25). It follows that E is a loop space, that is, there is a space Z with E = Z. Thus [W, E] is a group and we can identify co (X) as a coset of Hin (W; Fr). In particular, two liftings and : W E, differ by a mapping of W to K(/p, n + t — 1). (Recall the exactness of [W, F] [W, E] [W, B] for a fibration F E B.) It follows that 4),,o(x) determines elements fa o in Hm

(TV; Fp)/ct

o

(i . Hn+t -1 (147;

723))

where i* : [W, K (ElpZ,n+t —1)] —> [W, E] is (pre-)composition. As always, indeterminacies can be difficult to make explicit. In the best cases, dimensions conspire to make (1)0 : ker (W; 7p) a well-defined function. 2) This definition can be made for different coefficient groups and more general cohomology operations (IIMaunder641). We will not need this level of generality. 3) If V is a finite dimensional graded vector space over If, then we can write V= e 7p, where the nj correspond to the dimensions of basis elej=1 ments for V. Let K (V) = fi K(/p, n i ). Then, as graded vector spaces, j=1

A class in [W, K (V)] is representable as a vector (xl , . . . , x 8 ) Brij (W; 7p). Furthermore, the (abelian) addition on E/pE determines

7r*

(K (V))

in

ri

V.

a mapping, +: K(Z/pZ Z/pE, n) —>

which induces the vector addition on such spaces. We generalize the definition of secondary operations to vectors of classes. Let A: K(Vo) K (VI) represent an n-tuple of cohomology classes (x1, . • • , xn) E Hi (K (Vo); Fp ). Then we have the analogous diagram defining a secondary operation associated to A:

ei

K(s 1 171 )

E

K (71,1p27., m)

K(Vo )

For example, consider the Adem relation Sq2 S q 2 + Sq3 the mapping

= O. Let A denote

sq A = 2 ) : K(Vo) = K(/2, n) Sq

K(E/2E, n + 1) x K(7/2, n + 2) = K(Vi).

373

9.1. Motivation: What cohomology sees

By the naturality of the Leray-Serre spectral sequence we get a mapping of spectral sequences whose source is associated to the path-loop fibration over K(Vi) with target associated to the pullback of the path-loop fibration with respect to the mapping A. The class in the cohomology of the fibre Sq 3 z n Sq2 zn+ i E H* (K (s -1 V1);7 2)

goes to zero under the transgression in the target spectral sequence because the Adem relation holds.

N 't \

\ \

\

31

22

`4SqSq + SqSq )1, n

N t \

----\

N

/.

4 _.-----Sqin+i + Sq2ln+2

n+3,0 represents a class a E Hn+3(E; 72), The class [Sq3 z 7, se iri±i ] Eoo where E is the total space of the pullback over A. This gives a secondary operation as in the diagram:

W

A

> K(Vi).

If x E Hn (W; F2) and Sq l x = 0 and Sq 2 x = 0, then lifts ±- to E of x exist and (1),„A (a.) is defined. Furthermore, the indeterminacy is the subgroup SeHn (W; 72) + Sq 2 Hn±l (W; 72) of H 3 (W; 7 2). This example was employed by [Adem571 to show the nontriviality of 772 = Sn+2 —> S n where 77 denotes a suspension of the Hopf map 77: 83 —> 82 . On H* (C (n2 );7 2), the operation carries the generator in degree n to the generator in degree n + 3. We say that 7/ 2 is detected by

9. The Adams Spectral Sequence

374

the secondary cohomology operation a ,A. More generally, any mapping f: Sn±r -> Sn for which a secondary cohomology operation acts nontrivially on H* (C (f); 7p) is said to be detected by this operation and, in particular, f is not homotopic to the constant map. 4) The Adem relation Sq 3 Sql sq2 sq2 0 holds universally on the cohomology of any space. Any such quadratic relation between primary operations can be expressed as a composite, K (141) -> K (V2) -> K (V3), with x o e *. For example, in the case of the given Adem relation we have Cl q

K ( sny 2 )

S'qr2 )

■

K(s n+1 T2 sn+272)

+0 Sq 2)

> K(Sn+4 72).

Generally, we can construct a diagram of spaces on which to define the associated secondary operation: K(s -l Vi)

E

K(8 1 V2)

W Here E is the pullback of the path-loop fibration over K (VI) with respect to The mapping x, as a cohomology operation, produces classes in H* (E; 7,) because classes that transgress from the fibre of the fibration p are annihilated by x. If x: W -> K(Vo) represents a vector of classes in H* (VV; Fp) with ox s, then we obtain the secondary cohomology operation due to the relation x o * as a subset of [W, K(s -1 V2)]. 5)In order to capture all of the relations between primary operations we turn to a homological description. Let di

d2

) 0 H*(X; C0. C1 C2 be an exact sequence of A r -modules, with Co, C1 and C2 free A r -modules; Co can be taken as the free module on a set of A p -generators of H* (X; We can think of C1 as the free module on the Ar relations among the generators of H* (X ;7 p ) , and C2 as the free module on the secondary relations, that is, relations among the relations. If X is (n - 1)-connected and Ci Ap 0 Vi `=-'2n-1 H*(K(Vi ); Fp) for 14, a graded vector space, then we associate a diagram of spaces, where we have written the name of the A r -module map for a continuous map that induces it. (The identification of algebraic mappings : C -> C,_1 with d, : K(14_1) -> K(V,) follows from the representability of mod p cohomology.)

K (s - 1 Vi )

X

E

K (s -1 V2)

) K(Vo) — K(14)

K (V2) .

375

9.1. Motivation: What cohomology sees

Applying the mod p cohomology functor to the bottom row gives the sequence of Ar -modules, exact in dimensions less than 2n - 1,

(x; Fp)

Co

d2

d i ci

C2.

The secondary operation arising from this diagram lies in [X, K (s -1 V2)]. We say that this operation is the second order operation associated to Co <— (12

di

C2.

6) Paragraph 5) can be generalized to mth order cohomology operations. Continue the exact sequence of free A r -modules: 0 <- H* (X; F p )

CO

<-

C

• • • <-

Cm .

This gives rise to a tower of fibrations: K(s -(11-1) Vm_ i )

K ( s -2v2 )

Em _ i

-2 v3 ) 8-24 K -2 v4 )

E2

K(s 1 14)

El

X

K(V0)

In degrees less than

—,K(s -(712-1) Vm )

K (8 -1 V3)

K (s -1 1/-2)

di

K(Vi)

d2

K (V2).

r, the map in cohomology, induced by the composite -> K(s - (r -1 )Vr.), is s- (r-1 )d, . This follows from the Serre exact sequence (Example 5.D), applied inductively to each fibration. In order to get a lifting of r: X —> K(V0) to E: X -> Em _ i and so define an /nth order cohomology operation on [X, K(s -("1-1) V,)], it is necessary that the rth order operations defined by this tower, for r < m, all contain the zero class. Only then is a lift to the next stage possible. When f: St Sn is a mapping and X = n U f en±t ±1 , if the lifts of the class in Hn (X) determine only nonzero classes in Hn+t+ 1 (X), then we say that f is detected by an mth order operation and f is not homotopic to the constant map. 7)The dependence on the connectivity of X can be removed by considering only stable mth order operations. Such an operation 1 determines a commutative 2n -

376

9.

The Adams Spectral Sequence

diagram sHn(X;Fp )

sod,

sHn±t (X;7,)

—1 Hn±l (SX; 7p) M Hn±t±l (SX; 7p)

The details of the construction of these operations and their properties can be found in the work of [Maunder631. The connection between higher order stable cohomology operations and the spectral sequence of Theorem 9.1 lies in the interpretation of the filtration on ( p)rr!' and (){Y, X} to which the spectral sequence data converge. Proposition 9.2. If an element u G (p) can be detected by an nth order stable cohomology operation, then, for some m < n, u Fm(()), the mth stage of the filtration of (,)7 s associated to the Adams spectral sequence.

A proof of this proposition will emerge with the construe lion of the spectral sequence. The role of higher order operations in homotopy theory is fundamental and the Adams spectral sequence helps to codify and suggest their further use. The interested reader can consult the papers of [Cohen, R81] and [Lin761 for other possible applications. 9.2 More Homological Algebra; the Functor Ext Before we construct the spectral sequence, a digression into homological algebra is necessary to secure the algebraic tools. In this section we also construct a spectral sequence associated to an extension of Hopf algebras based on the change-of-rings spectral sequence of [Cartan-Eilenberg561 and elaborated by [Adams581. In Chapters 3 and 7, we studied the categories of modules and differential modules over rings and over differential algebras. The tool of choice was the functor Tor that measures the deviation from (left) exactness of the functor M OR —; ToriR (M, —) is the ith left derived ftinctor of M (SR One of the fundamental relations in homological algebra is the Hom-tensor interchange: When A is a left A-module, B is a right A-module and B and C are left F-modules HomA (A, Homr (B, 0) Homr (B

OA

A, C).

This isomorphism plays a key role in the Universal Coefficient theorems in topology. The functors Homr (M,—) and Homr (—, Ar) are half exact. We next study their derived functors. We begin by identifying the category of interest. Let (r, cp) denote a graded algebra, over a field k, with product cp. We generally assume that r has a unit,

9.2. More Homological Algebra; the Functor Ext

377

E: k —> r, as well as an augmentation, 17: r k (we assume 7/ O E = idk). The category of graded left F-modules is denoted by r Mod, and we take the morphisms in this category to be of degree zero. Denote the F-linear maps between two left F-modules, M and N by rMod(M, N). The suspension functor on r Mod is defined as follows: If M E r Mod, then sM is the graded vector space, (S M )71 - M71-1, with F-action given by -y • (sx) = ( 1)clegeys/(-y x), where x G Mn and sx is the corresponding element in (sM) ±1 . Define the iterated suspension by sn = s o 8n-1 and s 1 = s. The graded version of the Hom-functor is given by -

Homil, (AI , N) = r Mod(M, snN).

Equivalently, a r-module homomorphism in Hornil,(M, N) can be thought of as a homomorphism f : M —> N that lowers degree by n. To study the derived fiinctors of Homr* (M, ), we resolve a left r-module M by projective r-modules. That is, construct a long exact sequence in r Mod; —

d

0

M

d

<— P1 <— • • •

d

d

where each Pi is a projective module over F. (The reader should contrast this with the accounts in §2.4 and §7.1 where the presence of a differential is part of the construction.) Let N E rMod and apply Hom *r(—, N) to this sequence HOTri r * (Po , N)

d

Hom( Pi

,

N)

d

Homr* (Pri , N) L....

The homology of this sequence defines Ext r* ' * (M, N). We leave it to the reader to verify the usual properties of this derived functor: (1) The definition is independent of the choice of projective resolution. (2) Exer'* N) = Homi%(M , N). (3) Given a short exact sequence of left F-modules, O —> A B C 0, there are long exact sequences, —> Extr (C, N) —> Ext*(B, N) —> Extr (A, N) —> Extri±l '*(C, 1V) —> and e5 —>Ext * (M, —> Extrp* (M, B) . Extnr '* ,C) —> Ext rir+1'* (M, A) (4) Ext is functorial in each of its three variables. (5) Extnr ' * (P, M) = {0} if n> 0, P is a projective F-module, and M is any left F-module. Furthermore, Extnr ' * (M, J) = {0}, if n> 0 and J is an injective r-module and M is any left r-module. The notion of an injective module over F is the formal dual (mono replacing epi and arrows reversed) of the notion of a projective module. The key property of an injective module J is the exactness of the functor Homr (—, J).

9.

378

The Adams Spectral Sequence

We could have defined Extr' * (M, N) by forming an injective resolution of the module N and applying the fiinctor Homr (M, —). The Ext groups are bigraded. If we write Exts = Ext s '*, the single grading refers to homological degree and forgets the internal degree of the mapping. In the applications to follow, we focus on the computation of (X) = {S ° , X} * . Theorem 9.1 introduces a spectral sequence, converging to 0,)7s (X), with the E2-term given by Ext*.f (H* (X; 719), 7p). Anticipating these computations, we consider the computation of Extr"(M,k) and some convenient resolutions for computation. In the particular case M = k, we write

Extr (k, k) = H" (r),

and H* , * (r) is called the cohomology of the algebra F. The dual situation, given by Torr* * (k, k) = (r), defines the homology of the algebra F. Henceforth, we assume that the algebra r is of finite type over k (that is, in each degree n, rn is finite-dimensional over k). For graded vector spaces of finite type, the definition of the dual is straightforward: (Fdual )n = Horn k(rn k). Furthermore, if M' and M" are both of finite type, then so is iyual w dual 0k itri clual . M" and ( m l k m It follows that an algebra (F, v)) and a left F-module M, of finite type, with module structure map 1p r ®M .7u, yield by duality a coalgebra (rdua',(p*) and a comodule over the coalgebra, r dual , with structure map mdual

rdual ok mdual .

Recall from §7.1 the definition of the bar resolution:

olvi- <—rom<—ro.r(r)om<—ror(r)or(r)om<—•••, it4- and /(r) = ker(7): r where an (r, m) = F ® I(F)®' k). The differential and the contracting homotopy are given in Proposition 7.8. To compute Extr (M, k), we apply Hom(—, k) to the bar resolution. Since k, as a F-module, is the /(F)-trivial module concentrated in degree 0, it follows via the Hom-tensor interchange that Hor4(_,V, k) = Hornt (k or N, k) = ((k or N) dual )t .

The bar resolution becomes the sequence m dual

I(F)dual

mdual

i ( r )dual

i(o dual wdual

where 1/)* is the composite mdual '1)* _4',dual

md ual i*O1 /(F) dual

mdual

9.2. More Homological Algebra; the Functor Ext (i*

is the dual of /(r)

d* Gar

•••

(recall that V* (cti) =

an ]

= n,

379

r) and d* is given by

=

i)1+deg aa

for a, an element in a graded module) where rn a' ® a" and q/3* (A) = ai A;'. Thus /=l

Ext(M , k) = H (B(F)dual

Mdual , d . ),

where B(r)- = /(r)on. dual To compute the cohomology of the algebra r then, we can use B (r) dual, and differential which consists of elements [cti ' " a n ] with ai E /(r) d*(Ecti

I

•••

an]) =

1

—1 [61 '• • ai-1

I

I

•

I

•••

I an].

We point out that such a construction can be made with any augmented coalgebra, (C, A, n ), where A is the coproduct, J(C) = coker(n: k C) is the cokernel of the augmentation, which is the dual of /(Cd" a1 ). This functor on coalgebras is sometimes denoted F,,(C) and called the cobar construction on (C, A,17). It was first introduced by IAdams561 to compute 11* (0X) as a functor of the chains on X, C* (X), as a coalgebra. We can express the cohomology of r in terms of the cobar construction by H*,* (r) = H (phial) d.) . Another application of the Hom-tensor interchange shows the duality between the cohomology and homology of an algebra. Since we are over a field,

the Universal Coefficient theorem allows us to interchange the homology operator with Hum* and obtain the equation, when r is of finite type, Extr (k,k) = H(Homr(B(r), k)) = H(Homk(k = Homk(H(k er

Or B(r), k))

B(r)), k) = Torr (k ,

In the next section, the natural coalgebra structure on Tor (k, k) provides a natural product on Extr (k, k). Some calculations are eased by working in the dual. Finally, when speaking of the computation of Ext, we mention another computationally convenient type of resolution.

Definition 9.3. A homomorphism, : M N of left r-modules is said to be minimal if f (111) c r(r) • N. A projective resolution of a module M is said to be a minimal resolution if every mapping in the resolution is minimal.

9.

380

The Adams Spectral Sequence d

Proposition 9.4. Let 0

d

d

d

M Po <— P1 — P1<— P2 <— • • • be a minimal resolution of M by projective F -modules. Then Extl, (11 , k) Homr (P,, k).

PROOF: We assume everything in sight is of finite type. We exploit the duality between Ext and Tor for a proof. The duality in this case is between Extr (M, k) and Torr (k, M), and the dual statement of the proposition is that Torrs (k, M) k Or P,. However, for any P-module X, k Or X = x j(r) • x, which follows from the definition of k or X as X /

k

/ {(1 •-y)

x — 1 e ey • x)} ,

k or Ps —> where 1 •-y = 0 when •-y is in i(r). Since d(P,) c /(11 -Ps _ i , k Or Ps _1 is the zero homomorphism for every s > 1 and so Torrs (k, M) = k Or P,. Passing to the dual, we get a complex with all differentials zero and so Extl, (.111 , k) Hom r (P, , k).

Minimal resolutions come in handy for doing low-dimensional calculations or to begin an induction. As an exercise, the reader should compute H* , * (A (x)) where A(x) is the exterior algebra on one generator x, here taken to have odd degree. A minimal resolution or the bar construction can be applied to obtain H* , *(A(x)) k[y] as vector spaces, where y has bidegree (I, deg x). In the next section, the multiplicative properties of Ext are developed and we find that this isomorphism is true at the algebra level. Multiplicative structure on Ext

The bigraded7p -vector space Ext:i* (H* (X; 7,), H* (Y; 7,)) enjoys some further structure. There is a product when X = Y, and more generally, pairings of Ext groups. We give two constructions of the same operation. We will present the first construction in detail and sketch the second. The first identified by IYoneda541. Suppose that F is an algebra over a field k. Theorem 9.5. Let L, M, and N be left F-modules. Then there is a bilinear, associative pairing, called the composition product, defined for all p, t, g, t' > 0, o: ExtPt (L, M) Extr (M, N) —> Extf,±q,t±t. (L , N) .

PROOF: Let 0 L P. be a projective resolution of L, and 0 M Q. a projective resolution of M. If [f] lies in Ex-Opt (L, M) and [g] in Extr' (M, N), then [f] may be represented by f: P —> st M and [g] by g: Q, st' N . The following elementary facts about the suspension functor are left to the reader to prove: (I) If X is projective, then sX is projective. (Hint: s(r 0v) r ® sv for a graded vector space V.) (2) r Mod(sW, sX) (3) If 0 <— X <— W.

r Mod(W, X).

is a projective resolution of X, then 0 is a projective resolution of sX.

sX sW.

9.2. More Homological Algebra; the Functor Ext

381

We define [f] o [g] as follows: Using the defining property of projective modules, lift f: P —> stM up the resolution to f, : Pp+, —> st Q , for g > O. Suspend g to st g: st Q, —> st+t'N and let [f] o [g] = [st g o fq ] in Extrq't+t' (L, ) The following diagram depicts this construction: ,

PP-kg

If

q

s t' N

Because we can lift f, to fq+1 and g odm = 0, it follows from the equation stgo fg od r,= st gost dm o f, +, = 0 that stgo f, is a cycle. To show that all of the choices made in the construction are irrelevant, observe that two choices differ by a chain homotopy and so the difference vanishes on homology. The bilinearity and associativity are elementary to establish. We remark that speaking of the suspension is the same as speaking of maps that change degree and so, by introducing the appropriate signs, this entire discussion can be carried out without the suspension. We do this later. Yoneda's original construction of the composition pairing is useful both conceptually and computationally. The construction depends on the identification of M) with equivalence classes of exact sequences of r-modules of the form 0 —> M —> En _ —> En _ 2 —> • • • —> _Er()

L 0 (ri > 0).

Two exact sequences of this form are said be equivalent if there are homomorphisms (pi : g that commute with the morphisms in the sequences and the identity maps on L and M. Given classes in Extf, (L, M) and Extl,(M, N) we can take representative exact sequences, 0 —> —> Ep _ j —> Ep _ 2 —> • • • —> Ej —> Eo —> L —> 0

and

0 —> 1\7-

Fq _1 —>

Fq -2

—>

•••

Fl

O.

FO

To represent the product splice these two sequences together at

0

N

F,_ 1

• • • —> F0

E1 P-

•••

E0

M:

L

0.

[Yoneda54] showed that this pairing coincides with the composition product. An immediate consequence of this identification is the following result.

9.

382

The Adams Spectral Sequence

Proposition 9.6. If 0 —f A —f B —> C —> 0 is a short exact sequence of F-modules representing a class a inExt rl (C, A), then the coboundary maps in the long exact sequences derived from the sequence, 5: Exq, (A, N) —> Extr (C, N)

6: Exq, (N, —> Extr (N, A),

and

are given by left and right multiplication by

o,

respectively.

The internal degree of such an n-fold extension is given by the total change in degree from left to right. That the internal degree of a product is the sum of the internal degrees follows immediately. Corollary 9.7. Extr"(M, M) is a bigraded algebra over k with the composition product as multiplication. Furthermore, Extr" (L, M) is a right, and Extr" (M, L) a left, Extr (M, M)-module.

With a concrete resolution like the cobar construction on hand, it is reasonable to attempt to represent the composition product on Extr" (k, k) at the level of elements in the cobar resolution. Let [al • • • op ] be in Fp (r d"1 ) and k q ] be in Fq (rdual). Notice that the internal degree of a: I(F) [01 • • • is the degree on which a is nonzero, that is, deg a = t if a = 0 as a mapping i(r)t k. We define a product on F* (rdual ) by juxtaposition [at I

•••

I op]

[8 1

•••

/3q]

kti

'••

(-tp

/31

'••

/3q].

This mapping is clearly bilinear and associative. To see that it induces a product on Extr"(k, k), we show that the differential is a derivation. Let U = [cti • • • ap ] and V = [op+) • • • cep±q], then d* (UV) = d* Gal P+q

)--

jl

i=1

±V g /

•

Ct P±d)

[all • • •O jj [

•••

• • • ctp+qi

'••

api[ap+i

• • • ap+q]

[011 ' ' ' C-tp][t-tp+1 • • • CEp+r- 1 Ctpi ±r,u Ctp "+r,u • • • aP+q]

= (U)V + (-1)P-E t U d* (V),

where t is the internal degree of U, that is, t = EiP deg a i . Thus, this juxtaposition product induces a product on Extr" (k, k) with the correct bidegree.

9.2. More Homological Algebra; the Functor Ext

383

Theorem 9.8. The composition product and juxtaposition product coincide on Ext r" (k, k). PROOF: We show how to lift a given mapping through the bar construction. Suppose [f] Ex* (k, k) and [g] E Ex* (k, k). Consider the diagram where

(eY [71 ' • •

-Ys+i]) = 'T[T1 ' • • 'N]f ([7i+1 ' • •

_ r i(r) s+t f4

F®

r /(r) s-ki

_ r /(r) s fol

f4

/- (F)t

F®/(F)

r /(r) s—i NçN > k

>F

O.

gi This lifting satisfies d o f = L_, 0 d and so we have a mapping of part of the one resolution to the other. By the definition of the composition product [f] o [g] = [g o ft ]. The value of go ft on a typical class -y [-y' eYs+t] is When we represent [f] and [g] in the -rt+s]). I • • • f Œrt+1 I g(7[71 " ' 7t]) cobar complex as tensor products of elements of rd "1 we get exactly the value obtained by applying the juxtaposed dual elements to a typical argument and adjusting signs for the suspensions. Thus the products coincide. The simplicity of the product induced by juxtaposition allows one to do computations at the level of the cobar construction. This is especially useful for determining such secondary phenomena as Massey products (see §8.2) and —1-products. Suppose F is a Hopf algebra with cocommutative coproduct ti) and counit 11: F —> k. We assume further that r is of finite type over k. With these data, there is yet another way to induce a multiplication on Ext r"(k,k). Suppose we are given a projective resolution of k, X. k 0 with the homomorphism E : x0 =r—>k—>o, the counit of F. Let X 0 X be the complex with (X. 0 X.), = ED X j and differential d o (x y) = d(x) y + (-1)P+rx o d(y) (x (X p )r), then X. 0 X. can be given a F g F action via the twist map and we obtain a projective r 0 r resolution of k = k 0k k.

Using properties of the coproduct, counit and projective modules, we can construct a map, A : X. —> X.0 X., making the following diagram commute: X. A

<

k 0 k k < 60,

r0r,

x.

x.

384

9.

The Adams Spectral Sequence

Applying the fiinctor Hom *r ( , k) induces a product Homr (X., k)

Homr (X., k)

Homr (X., k)

that reduces to a product p: Extr (k, k) ®Ext rs' 't' (k, k)

Extrs±s' ,t+t' (k, k).

One can construct an explicit A on the cobar resolution from which it is easily seen that p is the same product as the one induced by juxtaposition. We introduce this construction to prove the following result. Theorem 9.9. If the coproduct on r is cocommutative, then the multiplication on Extr (k, k) is graded commutative with signs given by a

o

i)ss'+tt'o . a

for a E Extr (k,k) and 13 E

k).

PROOF: We extend the diagram in the construction to another row: 0,

k

0

ke k k

i

0

,

E°6

i kg k k<€®€

r<

X.

rer,

xi ox„,

Ti rer<

Ti xi ox.

Since r is cocommutative, = TO and A is chain homotopic to TA. This proves the theorem. On the cobar resolution with its juxtaposition product it is apparent that A and TA are not the same mapping. An explicit chain homotopy can be constructed. If we restrict our attention to k = 72, this chain homotopy allows us to define —i -products and hence Steenrod operations on H*,* (F). For k = 7p , a similar construction over the (p — 1)st iterate of the coproduct, '01) :r—>ro•••®r(p times), allows one to define the mod p Steenrod operations on H*,* (F). We refer the reader to [Adams581 and [Liulevicius62] for details of these constructions. The elementary properties of these operations are listed next for later applications.

9.2. More Homological Algebra; the Functor Ext

385

Theorem 9.10. Let (F, V), A, e, n) be a cocommutative Hopf algebra over IF. = 2. There are operations Sqi on Extr' * (72,72) that satisfy the following (a)p properties: (1)

: EXtr(72,72)

(2) Sqi (ab) = (3) ,S' qr

=

EXtrs+i'2t (

F2, 72 ) ,

(a)Sqn (b),

m±n=i [.9/ 2] (8 — t — 1)s r+s-t t q ,Sq ,

(4) Sq° ({[oe i ' • • an] }) = [cti ' • • ct,2,1}, (5) Sqsx = x 2 if x E Extl, (72, 72). (b)p, an odd prime. There are operations Pi and 13Pi on Extr (Fp , 7p) that satisfy the following properties: (1)

: Extrs' t (4, Fp) :

(ab) =

(3) Pr Ps =

Pr OP' =

(2i - t)(P -1 )+1 'rt

(7 p , 7p),

(b),

t>i>o i> j>0

/3P3 (a)P

2 (b) (—i) d eg a

Pi (a) Pi — (b)

((p — 1 )(s — t) — 1 ) pr+s-t pt r — pt [rho] i rt ((P — 1 )(s — t)) pr±s-t)pt t=o r — pt [r -1 /131 ((p — 1)(s — t) — 1) pr+s-t (pt), t+0 r — pt — 1

(r /P]

irkt

t=0

(4) P° ({[cti ' • • an]}) =

(5) Pr a

(2i-t)(p-1),pt

Ext rs't (Fp Hp) -> Ext;-,+

(2) Pi(ab) = 13P9

EXtr8+

['cti ' • • ala

=aP if a E Exq t (7p , 7p) and 2e = s + t.

These operations differ from the usual Steenrod operations because Sq ° and P° are not the identity mappings and Sq l and o are not Bockstein operators. The explicit expression for Sq ° on a class in the cobar construction will be useful later. We consider a simple example over k = 72: Suppose the Hopf algebra, F, is the divided power algebra on a single generator F(s). Recall that r(x) is generated as an algebra by generators -y, (x) for i = 0, 1, ... with -yo (x) = 1, -Y1 (s) = X and deg-y, (x) = i deg x. The product is determined by the relations ri( x ) -ri( x)=

-

and the coproduct is given on generators by 0(-yi(x)) =

i=

-yi ( x)

-yi _ (x).

386

9. The Adams Spectral Sequence

We choose this Hopf algebra to study because its dual is given by F(x) dual = IF 2 [y], where y is dual to x. The reader can easily read the Hopf algebra structure on k [y] from the product and coproduct on r(x) and it agrees with the usual polynomial multiplication and coproduct

'0(0) =

q.)Y i Y q—i • i=0 (

To compute H*,* (F (x)), we can apply the cobar construction to the coalgebra (72[y], ii)). The following lemma holds generally. Let F be a Hop f algebra of finite type over a field k. Then (rm. H 1, *(r) = Ext*(k,k) = Prim* (Fi)

Lemma 9.11.

PROOF: We recall that Prim*( ) is the functor that associates to a Hopf algebra its graded vector subspace (in fact, sub-Lie algebra) of primitive elements (see §4.4). Also the functor Q*() associates to a Hopf algebra its quotient vector space of indecomposable elements. When r is of finite type over k, prim* (pdaai ) Q*(r), so it suffices to compute Pr im* (F) dUal . This may be defined as the kernel of the reduced coproduct

ço * : i(rdual)

/(rdual)

/(rdual) .

From the cobar construction, this kernel is exactly Extrl ' * (k , k).

To obtain Hl, *(r(x)) then, we find the primitives in 72 [y]. The arithmetic of binomial coefficients (Lucas's Lemma, see §7.3) determines the primitive classes = [y 2 '] Ext r1 '"(',0 (k, k), for i = 0, 1,.... The products of these classes can be identified in the cobar construction, where they are cycles; /iii, for example, corresponds to [y2' ]. Coboundary formulas, such as 6 [Y3 ]

= [Y2 y] + [y

= 1 1 10 + loll ,

show that the product on the sub algebra determined by the generators {/i } is commutative. To complete the computation, one must show that the products of the li's are not boundaries and furthermore, that no other class can be a cycle. Again the arithmetic of ( q. i

mod 2 can be applied and one deduces that H" (F (x))

1F 2[1, i = 0, 1,

,

as bigraded algebras where each 1, has bidegree (1, 2i deg x). A corollary of this computation is the fact that Sq ° (1i) = 1,+1 and Sq l (1,) = l? in Ext ) (72, F2). The reader should provide any further details needed to feel comfortable with this computation. We will use the results in computing the cohomology of A2, the Steenrod algebra.

9.2. More Homological Algebra; the Functor Ext

387

A change-of-rings spectral sequence

We next introduce an analogue of the Lyndon-Hochschild-Serre spectral sequence (Theorem 8 bis .12) to compute H* , * (r), when r is an extension of Hopf algebras 0 —> A —> r—>rilA 0. Here 17/A = ro-(A)•r and A is nornaal in r, that is, /(A) • r = F • /(A). This spectral sequence can be derived additively by methods of [Cartan-Eilenberg561; its multiplicative properties were proved by lAdams601. Theorem 9.12. Lei A be a sub-Hopf algebra of a Hopf algebra F. Suppose A is central in r, that is, ab = (-1riega deg b 0 a for a G A, b E F. Then there is a spectral sequence, converging to H* (r), with Hg (A) Ok

HP(rfiA),

and differentials dr of bidegree (r,1 — r). Furthermore, this spectral sequence converges to H* (r) as an algebra with the product structure on the E2-term given in the isomorphism by (x y) • (x'

= (-

1)pq-Pte (xx/) g (yyr)

when y E HP ,t (17/A) and x' E Hq ,( ' (A). PROOF: Before beginning the proof, we observe that this spectral sequence is actually trigraded—the third grading is given by the internal grading t in H* , ' (). All of the differentials preserve this grading and so it is carried through to where it corresponds to the internal grading on H* ,- (r). This hidden grading plays a key role in computations (§9.6). The proof exploits duality and so we begin with B(r), the reduced bar construction on F. Filter B (F) by

FB(F) =

{

I

• • •

ys ] for which at least s — p of the -yi lie in /(A) 1.

-

This is an increasing filtration with F°B (1-') = B(A) and F8(F) = Since A is a sub-Hopf algebra, d(FPB(F)) C FPB(r). Thus we have a spectral sequence, converging to Torr (k, k) with El -term given by = Hp+,(FPB(111FP-1 B(F),

d° ).

To prove the theorem we first establish a chain equivalence between the E° term, (FPB(r)1FP– i-B(r), do), and (B(A) /(17/A)®P, d 0 1). B(A) /(17/A) 029 Consider the mapping, vP : FP (F) VP

OEY1 • • •

I -rs1) =

I

•••

I -rs—p]

7M/ s—p+1

0 • • • 0 7rys,

9. The Adams Spectral Sequence

388

where F —> 17/A is the projection. Since 7rA = 0 for A G /(A) and [-yj • • • ye] is in FP B (F) if at least s p of the -yi lie in /(A), vP • • • 7.91) = unless exactly yj through -ys _p lie in /(A). Hence /JP is well-defined. We show that the following diagram commutes 7:

—

B (A) 0 /(17/A)®P

F(F)

a

l

dO1

> B (A) 0 I(17/A) ® .

FP B (r)

Since vP is zero except on elements of the form [A1 • • • As p 7s 2+1 • •• -y s], we check that vP o d = (d 0 1) o v 2 on such an expression. We write —

I N-p-ki

d([Al • • • As—p

•••

Eis_ 113 1 [A l • • • +

[A l

=

A

•

I

• • • A 8 _2_,

-

-rs-p-ki •• •

A8-278-2+1

• • 17s]

P—' _

=1

[Al • • •

Observe that vPGA1 7r(A s _p -ys _p+j) = 0 vP (Pki

A8-2

7s—p+1

"Ys—p+j7s—p+j+1

**

=

• • • A 8 _2 _, As _2 -y8 _2+ 1 • • •

I 0

because

and that

As—p 'Ys—p+11 • • • N—p+j7s—p+j+1 • • • 1781) = [A11* • •

As-2-1]

® 7rAs-p 7ffys-p+i

by the definition of vP on B(r) 8-1,. . Finally, observe that vP takes FP"B(r) to mapping of complexes PP: (FPB (r)IFP-1-B (r),

{0}

e • • •

and so we get an induced

(B (A) 0 /(17/A) ®P, d

di)

0 77.9 =

1).

We can rewrite FPB(r)IFP— IB(r) as k

OA

A0

F(F)/F"-

(F)

and so we plot a circuitous route to showing PP is a chain equivalence by showing gP = 1

PP:

A

F(F)/F

1

(F) —>

A®

B(A) /(1//A)®

is a homology equivalence. We introduce some associated complexes: Let C (p) = A® FP ±F0FP-1 , where we write FP = FP B (F). Then the bar construction differential d, defined on B (r, r, k), takes C (p) to itself, as does the chain homotopy s. (You'll find

9.2. More Homological Algebra; the Functor Ext

389

the formulas in Proposition 7.8). It follows that C(p) is acyclic. There is a surjection C(p) —> A 0 FP/FP-1 that is induced by 1 pr where pr is projection. The kernel of this surjection is F 0 FP-1 . Increase p to p + 1 and the surjection C(p +1) » A 0 FP'/FP takes C(p) as a subspace of C(p +1) to O. This leads to a short exact sequence, which defines another complex K(p) as kernel, 0 —> K(p) —> C(p+ 1)1 C (p) —> A

FP+1 /FP —>

O.

The associated long exact sequence on homology implies Hq (K(p)) H q+1 (A

e FP-"-/FP),

because C(p + 1)1 C (p) is acyclic. We next analyze the complex K (p) that leads to the desired result via induction. Up to this point we have been using only part of our hypotheses—that A is a normal subalgebra of the algebra F. In the case of Hopf algebras we can apply the following remarkable consequence of the Hopf algebra structure due to 1Mi1nor-Moore651 (Exercise 6.12): Fact. Suppose A is a sub-Hopf algebra of r and A is normal in F. Then r has a basis as a A-module, consisting ofl and certain homogeneous elements in /(r); r is free as a right A-module on this basis. Furthermore, the basis projects to a vector space basis for 17/A.

We denote such a basis for r by {-yi } and its image in rllA by {coi = It follows that we can write r as a right A-module by

7*(i )}.

F (17/A) 0 A = A + /(F//A) 0 A. A 0 FP-F'/FP is r FP, K(p) may be Since the kernel of C(p 1) written as the image of the inclusion followed by a quotient: K (p) = im(r

FP C (p + 1) I C (p)).

Replacing our expression for r as a right A-module, we get K (p) = im(r 0 FP —> C(p + 1)1 C (p))

= im(A 0 FP ± /(17/A) 0 A 0

(A® FP+1 /(F//A) 0 A /(17/A)

A

FP/AØ FP + i(F//A) 0 A

FP/FP-1 .

We proceed by induction to prove the following assertion: (9.13)

H,(A 0 FP/FP-1)

/(rfin)0P, 1. 0,

if q = p, elsewhere.

FP-1 ))

9. The Adams Spectral Sequence

390

Consider the composite, denoted by fP, fP : A

FP/FP-1 —, 1(2)uv A® B(A) /(17/A)®P €(1- ) /(F//A)P

For p = 0, fo is simply the mapping A 0 B (A) k given by the augmentation. But this is an equivalence by the properties of the bar construction. Assume (9.13) for p, that is, H(fP): H(A FP I FP') —> I(171A)® 2 is an isomorphism. By the properties of the complex K (p), we have H (K (p))

H(/(17/A) 0

A

FP/FP-1 ),

where the differential on /(F//A) is zero. Thus we have H (K (p))

/(F//A) H(A

0 FP/FP-1 )

/(F//A)°P+ 1 .

Since Hq+1 (A FP+1 /FP) (K (p)), we have shown that the assertion (9.13) holds in case of p + 1, and hence for all p by induction. When we apply k eA — to gP , we obtain the desired chain equivalence between FPB(r)/FP —i- B(F) and B (A) Ø /(F//A)°P. The El - term of the spectral sequence is given by (B (A)

/(F//A)

, d 1)

TorA ,, (k,k)

/(F//A)°P

To compute the E2 -term of this spectral sequence, we introduce a chain mapping TiP : B (A) ei(rilA)0P FP/FP-' that acts as an inverse to PP and induces a chain equivalence on B(F//A). First, use the basis for F//A, {co 9 }, to split the projection 7 and obtain a map u: FllA r of graded vector spaces. FPB(r), Next, introduce a version of shuffle map, ti,P : B(A) /poi' given by AP Gai • • • ad0 [ctr+i • • • ar+p]) =

(-1) 6(') [a,( 1 ) • • • acrer+p)]-

(r,p)-shuffles o-

Recall that an (r,p)-shuffle is an (r p)-permutation that preserves order on 1 through r and on r +1 through r +p. The sign E( a) is the sign of the permutation (see the proof of Lemma 7.11). The key property of the shuffle product in this setting is the formula d( 1i (a 0 b)) = pP (d(a) b)) ±p13-1 (a d(b)), where a G B (A) and b E B(1'). This equation depends on the centrality of A in F. Let AP denote the composite TcP : B (A) 0 /(F//A)°P

100-P

B (A) 0 /(F)°P FPB(r) PP

F(F)/FI-(F).

9.2. More Homological Algebra; the Functor Ext

We leave it to the reader to verify that PP o Hq (B(A) /(17/A)®)

10d

= id.

>

391

Next consider the diagram

Hg (B (A) 0 /(17/A)P -1 ) _

Al" L1

L1

r,q

23-1 ,q•

For a E B (A) with d(a) = 0 and b E /(FllA) °P, 1

di g, (a 0 b) =

d ig (1 0 o- ) (a b)

=

ig -1 (Et

do-Pb)

= (1 ® irP-1 )(a, do-Pb) = d7rP o-Pb =ãø cb. Thus the diagram commutes and Ep2,q H, (A) 0 The last step in proving Theorem 9.12 is the construction of a coproduct structure on the spectral sequence (we are still in the dual spectral sequence for H ,(F)). Recall that B (F) has a natural coproduct given by //): B (F) —>

B(r) B(r), '0([-yi

I

•• •

I

I

=

•••

I

0 [-ri+i

I

•••

I -Ys]

This coproduct is dual to the juxtaposition product on the cobar construction. We filter B(r) B(r) with the tensor product filtration,

FP(B(r) 0 B(r)) =

P

i=o

F(F)

FP — iB(r).

One can check that

(1) 1/) is filtration-preserving and commutes with the differentials; (2) PP and t-i,P commute with gp if we equip B (A) and B (FfiA) with the same coproduct. (This requires that the shuffle product commute with which it does.) With these facts the isomorphisms yield that the coproduct on the E2 -term is isomorphic to the tensor product of the coproducts on II (A) and 11,, (rllA). Finally, to obtain Theorem 9.12, it suffices to dualize this proof. We have proved the dual of Theorem 9.12 for the homology of Hopf algebras. A consequence of the proof of the multiplicative properties of the spectral sequence is the analogue of Corollary 6.9 that the Steenrod operations on H* (A) and H* (17/A) commute with the transgression, dr : Er°,7-1 E'°. These

392

9.

The Adams Spectral Sequence

operations, for k =72, act classically on the homological degree but double the hidden, internal degree. To prove the analogue of Corollary 6.9, one must work directly with the cobar complex and the explicit chain homotopies that give rise to the Steenrod operations. The interested reader can consult the papers of 1Adams60] and 11-iulevicius621. With the necessary algebra all in place, we return to the problem of constructing the spectral sequence of Theorem 9.1. 9.3 The spectral sequence

The goal of the chapter is the computation of the groups, {Y, X}, where X and Y are spaces of finite type and Y is a finite dimensional CW-complex.

The tools of choice, in this exposition, are elementary; (1) the properties of fibrations, in particular, the construction of towers of principal fibrations and the exact sequences that result from these constructions, (2) the properties of the suspension, including Freudenthal's Theorem (Theorem 4.10) and (3) exact couples and their subsequent spectral sequences. We first construct certain towers of fibrations, known as Adams resolutions that realize geometrically an algebraic resolution of the modp cohomology of a space. The reader can compare these constructions with the Postnikov tower of a space (§4.3 and §6.1) and the towers built in §9.1 to describe higher order cohomology operations. By building such a resolution for a space X, then SX, then S 2X, and so forth, the mod p cohomology of the Adams resolutions assembles into better and better approximations of a free Ap -module resolution of H* (X;7 2 ). The application of the functor [Sm Y, —] to the system of fibrations results in an exact couple and a spectral sequence. Our analysis of the construction allows us to identify the E2 -term. We remark that the bigrading on the spectral sequence that results from the exact couple is not standard and, in fact, does not conform to the definitions in Chapter 2. This could be avoided by an unnatural regrading of everything in sight, but this would complicate matters further. The nonstandard grading does not affect the arguments that follow but might seem odd on first exposure. Next the properties of Adams resolutions are developed to demonstrate the convergence of the spectral sequence. Then a geometric pairing is defined at the El-level that provides us with a spectral sequence of algebras. The pairing is so defined as to yield the Yoneda multiplication at the E2-term. For the reader who has little interest in the geometric origins of the Adams spectral sequence, but interest in the computations, we suggest you skip on to §9.5 where the homological algebra of §9.2, the Steenrod algebra and the spectral sequence are used to compute some of the stable homotopy groups of spheres.

9.3.

393

The spectral sequence

The construction: Adams resolutions

Let X be a space (of the homotopy type of a CW-complex) that is of finite type over 1F„ and suppose X is (n — 1)-connected. The basic goal of an Adams resolution is the geometric realization of a free Ap -module resolution of H* (X; 7p) through dimensions n < t < 2n (the stable range). The first step is to choose a graded vector space o VP such that oVo

fl * (X ; 7 p)i I (A p) ' * (X ; 19P)

By the representability of the mod p cohomology fiinctor we can choose a mapping 0 F0 : X —> K(0V0) such that Vo)*: H* (K (0V0); p) H * (X ; IFp ) is an epimorphism in degrees less than 2n realizing the isomorphism when composed with the quotient. Having chosen the mapping 0F0, form the pullback of the path-loop fibration over it: OXi -, PK(OVO)

Consider the long exact sequence on cohomology (Example 5.D), for t < 2n-1, ( ;7p ): where we write ( ) for the cohomology Ht — l(X)

Ht — l(0 X 1 )

Ht—i (0K ( o vo ))

Ht (x) onto

Ht

Vu ))

The epimorphism, 0F0: Ht (K(0 V0)) —> Ht (X), is the transgression associated to the fibration o Xi —> X. Thus, for t < 2n — 1, Ht(0X1) maps isomorphically onto the kernel of (0F0)*. Observe, also, that 0X1 is at least (n — 1)-connected, and the map Ht (X) —> Ht (oX i ) is null for t < 2n— 1. Iterate this procedure for 0X1; that is, choose ()V) isomorphic in degrees less than 2n — 1 to H*(0X 1 )//(A p ) • ii*(0 X1 ) and a mapping 0 F1 : 0 X1 realizing an epimorphism on mod p cohomology. Pullback the pathK(014) loop fibration over K(0V1 ) with respect to 0F1 to get a fibration 0X2 —> oXi• oXi+, —> oXi —> • • • —> Continuing, we get a tower of fibrations, • • • oXi —> X, with each oXi at least (n — 1)-connected, and the mappings (oX i ) —> Ht (oX i+i) null in degrees t < 2n — 1;

394

9. The Adams Spectral Sequence

oXt-ki'F => P1 K(oVt+i) o x,

K(ovi)

K(0 V)

x

K(0 Vo)

The mappings S2K (0 14) —> 0 X i+1 properties. On cohomology we have F +1)* H* (K (oVi+i))

>

oFi+i

K ( 0 Vi+i)

(0 X) —>

have additional

If * (QK() 14))

where the first map is onto in degrees less than 2n and the second map can be taken to be a degree 1 map that is one-one, onto the kernel of Hs+'(K(0 14)) —> H +1 (0X,) in degrees less than 2n — 1. If we compose and desuspend, the homomorphisms H* (K (014+1)) —> s -1 H* (K (014)) can be assembled into a complex: 0 <— H* (X)

H*(K( 0 V0 ))

s -1 H* (K (0V1 ))

s -2 (K (oV2))

which, in degrees less than 2n, is a free ,Ap -module resolution of H* (X). The next step in this process is to repeat the previous construction based on X for SX, the suspension of X. Recall that H* (S X) sH* (X) as modules over A. Since SX is n-connected, the stable range extends to degree 2n + 1. We can relate the data from the Adams resolution of X to that of SX by choosing the graded vector spaces 1 V, to be s(0 V) EDW, where 144 is the additional term needed to obtain an epimorphism up to degree 2n + I. With these details of construction, the long exact sequences of cohomology vector spaces can be examined to show that 1 Xi and S(0X) have the same mod p cohomology in degrees less than 2n — j. This establishes the inductive step. We continue by building Adams resolutions over S2 X, 83X, and so on. Over SmX, we have a tower of principal

395

9.3. The spectral sequence

fibrations:

mX

K(V)

Xi

mF1 > K(V)

Sm X

.mF0 ) K( m V0).

7n

The properties of the construction can be summarized:

(1) each m Xi+1 —> mXi is a fibration with fibre QK(,Iti ), (2) each ri Xi is at least (n + m)-connected, (3) H* (m X i ;7,,) H*(S( m _ i X i);7,,) in degrees less than n + m — (4) H* (m X i ;7p ) > H* (ni X i+1; Tp) is null in degrees less than 2(n + Finally, desuspending the appropriate number of times leads to the complex, a free Ar-module resolution in degrees less than 2(n + m): 0

m H* (Sn I X) <— m H* (K ( m V0))

s — m -1 H* (K ( m Vi)) ‹— • • • .

We next prove an important property holds for Adams resolutions that is analogous to the universal property enjoyed by projective resolution. Lemma 9.14. Suppose {Wi —> Y, Wi +1 i > 1} is a tower of principal fibrations and there is an integer N > 0 so that, if S2K(Aii ) is the fibre of Wi+1 then Mi is a graded vector space, trivial in degrees greater than N. If X is (n — 1)-connected, 2n — 1 > N, and f: X —> Y is a continuous mapping, then there is a sequence of mappings, oXi for each i, such that the following diagram commutes,

I PROOF:

If

We construct the mapping from oX i to W1 and leave the inductive

9.

396

The Adams Spectral Sequence

step (essentially the same argument) to the reader. Consider the diagram

PK(Mo )

oXi TV) X

Y

K(M 0 )

Since Ht (X) —> H t (0X1) is null in degrees less than 2n — 1, the composite oXi K(Mo) is null-homotopic and so there is a lift of the composite to PK(M0 ). By the universal property of the pullback, W1 —> Y, we get a mapping 0X1 —> W1 , making the diagram commute. Corollary 9.15. Given two Adams resolutions of a space X, there are maps between them covering the identity. We finally derive the spectral sequence. First, form all of the Adams resolutions over X, SX, S 2 X,.... From the observation that each tower yields a certain part of an A r -free resolution of H* (X), let

0

H* (X; Fp )

Po

P1 < P2 <- • • • denote the limit of these approximations, which can be realized as —

= lim s — m —t H*(K( m -1/;)). rn—>co

For a given m, the Adams resolution of SmX yields the system of fibrations:

S22K( m Vi )

S2 (mX2)

S2K( m V2)

m X3

K(7073)

S22K( m V0)

S2(7,,X1)

K(,,V1)

7,7(2

K(7072)

52(Sm X)

QK ( m V0 )

mX i

K( m 171)

Sm X

K( m V0 ).

To this system of spaces and maps, apply the functor [SmY, —], where Y is a finite dimensional CW-complex. This yields an exact couple: [Sm+PY, ,X q ]

[Sm+PY, rri Xq_

[Sm+P-1 Y,,,X q]

[Sm+PY, K( m Vg_ i )]

397

9.3. The spectral sequence

(7,,X,_ 1 ), the p-fold loops on the libration; i is induced by SF (rnX,) j is induced by OP („Aq _i) —> OPK( m Vq _i), the p-fold loops on the classifying map of the fibration, Tri Fq _i; and k is induced by 52P K(-nVq -1) —>

Sr' (mX,), the (p — 1)-fold loops on the inclusion of the fibre. We have identified [T, çr ui with [SrT,U]. To fix the bigrading, let the first degree denote the level in the Adams resolution where the map is found, and the second degree denotes the codegree for the number of suspensions. This yields [Sm+PY, ,X,] = DQP ±q

[Sm+PY, K ( m Vq )] = Eq'P±g

The bidegrees of i, j and k are (-1, —1), (0,0) and (1, —1), respectively. When we display the unrolled exact couple, we get —> D8 ' t

)

Es

-

Ds- 1 ,t-1

D 1 4 -8+1

i

Et t—s+1

14-1

)

DO, t — S

E°4 '

,

Let X be In — 1)-connected, Y of dimension N and suppose that N m p < 2(n m). By the Freudenthal suspension theorem, [Sm+PY, SmX] [Sm+p+r+ly, rn+i xd , if < r. {Y, X} p . Furthermore, [sm+p-)ry, m xi]

So, we may write the groups, Ds .t and Es't as independent of m when we chose m large enough for a given s and t. In particular, when m is large enough, D" — s = [sm-Ft—sY, grn X

]

=

{ } , Al t- 8

We really have a spectral sequence for each m, but we think of the spectral sequences approaching a limiting value as m grows larger. We next apply the properties of Adams resolutions to identify the E2 -term that arises from this construction. In the system of librations, the mappings j and k arise from the inclusion of the fibre and the classifying map for the K (flys±i N) If we apply the Qt—i (ni x s+i) fibration Stt K (mV,) functor, [SmY, —], then the first differential is given by koj = d1 : Es 't+s =[Sm+t Y, K ( m Vs )] —> [Sm+t-1 Y, K ( m Vs+i)]=E9-1 't +s

However, for m large enough, [Sm±t Y, K ( m Vs )] = Hom?4, (H* (K ( m Vs )), H* (Sm±t Y)) = HomtAp (H* (K (7n Vs )), H* (SmY))

= HomtA, (s' P8 , sm 11 8 (Y)) = HomtA p (Ps , H* (Y)).

398

9. The Adams Spectral Sequence

Also, the mapping dl is simply Hom°A, (H* (K ( m Vs )), sm+t H* (Y)) nom(o.1) Honi°A, (sH* (K ( m Vs+i)), +1. H* (Y)) K( m Vs+1), which in turn where 0 is induced by S2K(rn Vs ) —> ,X s+i induces a: P 1 _> P,. Thus the following diagram commutes (up to the sign introduced by the suspensions):

Es,t+s

Es+i,t+s

Hom (P8 , H8 (AT )) 11°r11)' Homt.42 (Ps+i, 118 (AT )). This proves E2s ' t Ext i s (H* (X), H* (Y)) in the spectral sequence resulting from the exact couple and the first part of Theorem 9.1. Convergence The exact couple that gives rise to the spectral sequence has the "rightmost column" of groups, obtained by applying [SmY, —] to the system of fibrations, given by D° ,'' = {}', Xl t _ s . Unrolling the exact couple, we have a sequence of maps: —

Ds i,t+i

>

—

DO,t

—

s {Y; X } t s —

We now apply the methods of Chapter 3 to determine the convergence of the associated spectral sequence. Lemma 9.16. d,

Ers,t

PROOF: We can factor dr by dr •• Er

k(r) —> Ds+" — i mps-E n t-Fr-1

Filter {Y, X } 8 by Fs {Y,

Es+r,t+r-1.

X}, = imis : Ds'q±s —> D" = {Y, X}q.

399

9.3. The spectral sequence

Lemma 9.17. The filtration of {Y, X } * arising from a system of Adams resolutions does not depend on the choice of Adams resolutions.

PROOF: Apply Corollary 9.15 to cover the identity map SmX —> Sr"X between Adams resolutions. This system of mappings shows that each filtration is contained in the other. We can relate this filtration to the spectral sequence by using Corollary 2.10, which yields a short exact sequence, for each r, 13

Ds'* ker(i r : Ds,* —> Ds—r,*) ips+1,* ii

D8+r+1,*

Ds+i„ ) n ker(i : Ds -"-*

a")

0.

Let r go to infinity and observe that the left hand term of the short exact sequence

stabilizes when r = s, since j 8 : Ds , * —> D° , *. Lemma 9.18. There are monomorphisms

0 —> Fs {}, XIV Fs+1{-y, x} q —> Eq

PROOF: It suffices to show that F8 {} X}, I F,±1{ A-

,

-, Tq

Ds,*

: D8,8 _> 130, *) i.13.9 ±1, * •

Notice Fs {Y, X}, = isDs , q+s and F 8+ 1 {Y, X}, = i(i 8 D 8 + 1, q+ 8 + 1 ). There are short exact sequences 0

ker is + iD 8 +1, *

Ds,*

is+1D s+1.*

i s D s' *

D8 ' * /ker is + iD 8 +1, *

)

Fs /Fs+1

)

0

These maps are onto by the Five-lemma. We show that 1 is also a monomorphism. Let [a], [b] lie in D 8, */ ker is +iDs+ 1, *. If Ts [a] = [b], then is [a— b] = (a — b)] = 0. This implies either e (a — b) = 0, and so a b (ken or is (a — b) lies in i 8 + 1 D 8 + 1, *, which implies (a — b) lies in iD 8 +1, * . In both of these cases [a] — From Lemma 9.18, we have the exact sequence; 0 —> FIY, X1*IF8+1{y, x} *

—

> El*

-nr,m(ir: D8±r±1,* —> D 8 + 1,

)

n ker(i : D 8 + 1,*

Ds , *) —> 0.

To complete our proof of Theorem 9.1, we must show that the right term of this short exact sequence is trivial. We first develop some properties of the filtration.

400

Lemma 9.19. Fri {Y, X } q .

9. The Adams Spectral Sequence If a is in {Y, X} q and a is divisible by if, then a is in

PROOF: The term 'divisible by pn' makes sense in an abelian group, that is, there is a 9 in {Y, X}q so that a = pr/3. We proceed by induction on n for all spaces. The result for n = 0 is trivial. The case of n = 1 follows by observing that if f: Sm±gY —> SX is such that p f *, then the composite Sm+gY Sm X —> K ( m V0 ) is nullhomotopic since it represents classes in mod p cohomology. Therefore f lifts to „Xi and is in Fl {Y, Assume the result for n — 1. Since a = pfl = pn-1 a), we have that p lies in F1 {Y, X},, that is, p,(3 = iu where j: m Xi—> SmX and u: Sm-HY —> rri Xi . However, a = (0) = pri-1 (iu) = u), and so pn — u lies in Fri'{SmY, ni X i } q . Thus there is a map, w: Sm -HY —> r„X„ such that = p l u. Thus a = i(in'te) = irru)and a lies in Fri {Y X} q .

9.20. If ct is in {Y, X} g and a is not divisible by pri, then a is not in Fs {Y, X},, for some s. PROOF: We introduce an auxiliary space constructed as follows: In {X, X } 0 consider the element given by pri times the identity. Let h: Sm X —> Sm X represent this map. Pullback the path-loop fibration over S(h) to obtain the space U: Qs-4'x Qs7n±'_x-

PSm+1 X S m+1X

S(h)

smx

By construction, if X is n-connected, then U is (n + m + 1)-connected. If W is a CW-complex of dimension less than 2(n + m + 1), then by the Freudenthal suspension theorem (Theorem 4.10) [W, S2S 712+1 X] [W, Sm X]. We also have the exact sequence

[Tv, smx]

[w, smx]

[Tv, (I]

[Tv, sm -,1-x]

[147, sm+

,

which follows from the long exact sequence for the fibration. This traps [W, IT] between the coker pn and kerpn in the short exact sequence: 0 —> cokerpri —> [W,

—> kerpri —> 0.

Since pn annihilates both ends of this extension, [W, U] is a Z /p2-module.

9.3.

401

The spectral sequence

Let j be the first nonzero dimension with 7r7 (U) {0} (j > n + m + 1). If j < 2(n + m + 1), then 7r(U) is a 2 /p2n 2 -module by the argument above. Consider the quotient map, .71-i (U) )7r (U) (U) = 7r1 (U) Z/pE. Since ir1(U) /7,1pZ is a vector space over F, we can realize this quotient map by U —> K(7r1 (U) 0 7Z/pZ, j). Let U1 be the total space of the principal fibration induced by this map: ) PK (7ri (U) 0 2/p, j)

K (7r (U) O

The short exact sequence, 0 —> 7r1 (U) ) 7r1 (U) —> 7r1 (U) O Z/pZ —> 0, implies that 7r.) (U)) is piri(U). But 7r(U) is a Z /p2 -module, so if we repeat this procedure enough times, we get U 1 with 7r1 (Ui i ) = {0 } . Starting on 71-3±1 (U, 1 ), we can iterate the procedure until we eventually get to U' with irk (U') = {0} for 0 < k < 2(n+m+1). The tower of fibrations {Ui satisfies the conditions of Lemma 9.14 and so we have a mapping over the inclusion SmX X U:

Sm±g Y

Sm X

inc

U.

Recall a is not divisible by pr' and suppose a E Fs {Y, X}q for all s < r. Then a = irv for : Sm-HY —> ,X r . This factors through [Sm+gY,U] = {0} for Sm+QT in the stable range and so a is in ker (Inc) = [Sm±gY, Sm X] • But then a = pn 0, contradicting our assumption. Therefore, a E Fs {Y, X} q for some s < r. Corollary 9.21. F'{Y, )( }A, =fl FrilY, Xl q = {elements of finite order prime to p in {Y, X} J.

Notice that the assumptions that X is of finite type and Y of finite dimension play a role in the corollary. By Proposition 5.17 and induction over skeleta, we know that {Y, X }q is finitely generated and so there are no elements in the group of infinite divisibility by p.

402

9. The Adams Spectral

Sequence

Corollary 9.21 points out, however, that our filtration of {Y, X}, is not Hausdorff (§3.1) and so the spectral sequence cannot be expected to have {Y, X } „ as its target—after all, all our constructions were done mod p. We determine the actual target as follows: For an abelian group G let (r)G be the p component of G, that is, the quotient of G by the subgroup of elements of finite order prime to p. Then (p) {Y, X } s = {Y, X}„ I F" {Y, X}.. We induce a filtration on the p-component, (,){Y, X},, by -

c

Fs {Y , X}. I Fo 0 {y, x} C • • • C F 2 {Y , X}. I F.0 {y, x} c F i {Y,

Fo0 ty, xj. * C

&JAY,

X}8.

This filtration is exhaustive and convergent (Hausdorff), that is,

Us

! FQ°= ( 3){ }7, X }5 ,

n n FsIF" =

Furthermore we still have monomorphisms —> F8 Fcx- I FR+1 Fc>o

0

To complete our discussion of convergence, we prove the following lemma.

Lemma 9.22. Ecse PROOF:

F5 ( 3){ Y ; X} * I Fs+1 (0{Y, X * }

It suffices to show that

n(im

1

Ds+1,* )

D8+r+1,*

•

n [ker(i: Ds± l '* —> D")] = {0}.

Consider the relevant piece of one of the towers; (rn Vs )

mXs+n+1

m X,±1

m

X,

For a finite complex Z we have the exact sequence [Z, OK(m Vs )]

[Z,,X, ±{ ]

[Z,,X 8 1

[Z, K ( m V,)]

for which ker j = im[Z, aK( m Vs )] and p annihilates m Vs . Therefore, ker j is p-torsion. Apply [Z,—] again to the tower, and we can filter [Z, ,X, ±1] by the images of the in. But im in = F"[Z, m X, +i] and the argument of Lemma 9.20 carries over to show that in, in contains only elements of finite order prime im i n ker i = {0} since an element in a finitely generated top. Therefore, abelian group cannot be p-torsion and have finite order prime to p unless it is zero.

nn nr

an

,

9.3. The spectral sequence

403

Corollary 9.23. There is a spectral sequence, converging to (,)7 s , the pcomponents of the stable hornotopy groups of spheres, with E2 -term given by E2s Ht (A) = Ext tp (IE plF p).

In §9.5 we explore the consequences of Corollary 9.23 and compute some of the groups 0,4 for p = 2 and p = 3. Multiplicative structure on the spectral sequence

The multiplicative structure on a spectral sequence is often pivotal in the computations. We next introduce the composition pairing, o: {Y, {X, Alt —> {X, Z}, +t, that is reflected in a pairing of the spectral sequences converging to these groups. We sketch how the pairing at each level arises. Complete proofs of the existence and properties of the pairing are given in detail by [Douady58] or [Moss681 where different tools are to hand. We identify this pairing at the E2-term as the Yoneda composition pairing. When X = Y = Z, the pairing becomes a product and so we have a product structure on the relevant spectral sequence. Definition 9.24. Suppose a E Y}, and /3 E {Y, Z} t and suppose that f: Sm+s X —> SnlY and g: Sn+t Y —> SZ represent a and 0, respectively. Define the composition product of a and )3, /30 a to be the class in {X, Z}.+t given by sn+m+s+tx

>

sm+n+ty

Sr' g

sn+rnz.

Proposition 9.25. The composition product is bilinear, associative andfunctorial. The composition product induces the structure of a ring on {X, X}, and furthermore, {Y, X}, is a left {X, X},-module and a right {Y Y}-module. In fact, {Y, X},, has the structure of a {X, X},– {Y, Y } ,.-biniodule.

PROOF: These properties follow directly from the analogous properties of the unstable composition product, o: [V, W] x [U, V]—[U, W]. In particular, the pairings [V, W] x [SU, V] 1>[SU,W] ' V] —> [SV, W] x [SU, SV][SU,W]

[SV,W] x [U,

are additive in the second and first factors, respectively. (For the reader who is unfamiliar with these properties, we suggest Chapter 3 of the classic book of [Whitehead, GW781.) In the particular case of X = S ° , the ring structure on {S° , S° } = has better properties-7,,s is graded commutative. This follows by comparing

404

9. The Adams Spectral Sequence

the composition product with the smash product: (-1)(q-kotE,+/

sP

SP +k A 8q+1

sq+i

aA

EP-4d.

(_ i )

EPO

k Eq a

S P±k A Sq

) SP A Sq.

We have (— i)kEp+ko o Eq a _ 1)(q+1)k Eq +1 a 0 Eq 13. This result was first proved by [Barratt-Hilton53]. The relationship between this smash product and the E2-term, Ext, is through the external tensor product on Ext as defined by [Cartan-Eilenberg56]. We refer the reader to the blue book of [Adams74] for a thorough treatment of products. Let X be a space and construct a system of Adams resolutions for X. We construct 'pairings' on D" and E*'* of the resultant exact couple, that agree with the composition pairing on {X, X } * . Suppose [f] is a class in

D s,t

[stn+t—s-ke—s' x

and [g] is a class in Ds' ,e = [Sm+e— s'X, through the Adams resolution:

tn+t , —.9, Xs]

ra Xs , ].

By Lemma 9.14 we can lift

g

sm+t—s+e—s' x

m+V—s' Xs

1 1 1

m+e-s'X8-1

m+e- s' X1

stn+t —s' x

mXs'+s

1 1 1

gs

mX3'+.9-1

gl

g

mX3'+1

mxs ,

We define [g] o [f] = {[gs o f] for all choices of gs } c Ds±s' ,t ±t ' . Notice that if s = s' = 0, then this is the composition product on {X, X} * . Similarly we define such a 'pairing' on E** ; let [f ]

E s,t

[son+t-s-ke -s' x K( m+t,

and [g] E Es' 't' = [Sm+e— s' X, K( m Vs ,)].

Because the KW-1's are generalized Eilenberg-Mac Lane spaces and because X —> K( m+t,,,170 ) induces an epimorphism on mod p cohomology, Sm±t '

9.3.

405

The spectral sequence

we can define a mapping k0: K ( m +e -,, V0) —> K ( m 1/0 so that the following diagram commutes: m + — X1

gi

>

sm-kt —s'

„As , +1

m

K (n+t - s'Vo)

K (m 1/8')

The mapping gi exists since the composite ko

m+t' -.9' X1 —> S m*Ft' 8 ' X

K ( 77,17 )

K (m+e

is null homotopic. Inductively, if g +1 and k i exist, then we can find gi+2 an d k+i by choosing ki + 1 so that the following diagram commutes Qki

(rn+e —s'Vi)

(9.26)

g-F1

m±t'—s'Xi+1

1

mXs +i+1

\

1

1 , , ,

K (m-ke -eVi+i) —>. -tAT ,,,, v s, +i+i)

and choosing an appropriate lifting of gi+i . We can define the composition of [g] and [f] as [g] o [f] = llk, o f] all choices of k5 1 c E9 + 9 ' ,t +t '. We make some observations. 1. The choices made in all of the constructions differ by elements in the groups, [SqX, F], of homotopy classes of mappings to the fibres of the fibrations in the towers. If [f] and [g] are in Es ,t and Es' ,t ' and they are cycles under d1 , then their product [g] o [f] is also made up of cycles in Es+ s' ,t± t' and the differences vanish as an element in _W,t+t' Thus the defined 'pairing' is an actual pairing on E,* '* . II. When we apply mod p cohomology to the diagrams (9.26), and compute the effect of the maps k, then the construction is seen to be a geometric realization of the Yoneda composition product and, in the isomorphism of Er with Ext" A, (H* (X), H- (Y)), the products go over isomorphically.

406

9.

The Adams Spectral Sequence

III. The composition product is filtration-preserving on {X, X}.. To see this, examine the commutative diagram: sN±t±t'jc-

S N -Ft' X

gq + 1

.ggr

SN

V

NX1` -

This shows that o: FP {X, X} t,

NX q

1

N+V X P

N+t' X1

gr+q

NXq+ 1

IT N-

p+

Fq {X, X} t —> FP+q{X, X}t+t , •

IV. With some change in notation, we could have defined the 'pairings' for o: {Y, Z}. 0 {X, Y},, —> {X, Z},, just as easily. We now state the full theorem on products in the Adams spectral sequence, due to lAdams581 and [Moss68]. Theorem 9.27. There exist associative, bilinear pairings, functorial in the spaces X, Y, Z, all finite dimensional CW-complexes, and, for r > 2, Est (Y, Z) ®

't' (X, Y) —> E,8

' ,t +t ' (X, Z)

such that (1) for r = 2, the pairing agrees with the Yoneda composition pairing

Exe.4 (H* (Z), H* (Y)) Ø Ex-C:1'Ft' (H* (Y), H* (X)) —> ExC' A±: 't±t' (H* (Z), H* (X)).

(2) The differentials, dr are derivations with respect to these pairings, that is, in Er* , * (X, Z), dr (wv) = (dru)v + (-1)t-8 u (dre) q bidegu =

(s, t). (3) The pairings commute with the isomorphisms Er+i H(Er ,dr). (4) The pairings converge to the composition pairing o: {Y,

0 {X, Y}, —> {X, Z}, ,

that is, this pairing is filtration-preserving and the induced pairing on the associated bigraded modules is isomorphic to the pairing on the Eco -terms of the spectral sequences.

9.4.

407

Other geometric applications

Of course, these pairings not only allow us to compute more easily but they also allow us to define Massey products in the spectral sequence. We take up this notion in §9.5. The simplest computation reveals the role of the composition pairing. For any prime, p, the beginning of a minimal resolution of IF over Ap may be presented as in the diagram:

Pi Oti

o 0

E/pE.<— 72/pE

0<

Z/pZ <

2

,1

Ap

We identify an element ao in Ext 14 1 (Fr , Fp ) generated by the dual of the Bockstein, and occurring in this resolution as the element i Because the Bockstein has degree 1, we show that it detects the mapping p: S —> Sn, the degree p map. Consider the complex Sn Up 0+1 , the mod p Moore space, and the sequence pinch Sn Sn U

en+1

On cohomology, we have the extension 0 —> (5' 1 ) —> i/*(Sn

Up On —> fr(Sn) —> 0.

Since ii* (Sn Up 0+ 1 ) = Fp {x n , Oxn } is a nontrivial module over A p , this sequence identifies the only possible extension that can represent ao in Ext

(F2

7p ).

Suppose 1 E Exe; (7p , 72 ) is a nonzero permanent cycle in the Adams spectral sequence, that is, 1 lives to E. Also suppose a0 1 is nonzero. If 1 represents À in (p) 7riS then, by Theorem 9.27, ao/ represents the composition of 1 with the element that detects the degree p mapping. We conclude that pA 0 in (p)7ris s .

Much more is known about the product structure on 7 s . In a classic paper studied the global properties of the composition product. Using the extended power construction on a space and work of [Kahn-Priddy78], he proved a conjecture of Barratt that any element in 7r;:5 for i > 0 is nilpotent. [Nishida73]

9.4

Other geometric applications

The workhorse pulling the Adams spectral sequence along is the notion of an Adams resolutions together with convergence of the associated spectral

408

9.

The Adams Spectral Sequence

sequence. The construction is applied to a sequence of spaces X, SX, S2X, . The resolution for Sm X is obtained through a range of dimensions dependent on the connectivity of X. Taking these ideas as basic, we can enlarge the compass of application of the Adams spectral sequence to sequences of spaces with similar properties for which we can carry out the construction of an Adams resolution and identify the E2-term of the associated spectral sequence. Definition 9.28. A sequence of spaces X = {X1, X2,... X,. .} constitutes a spectrum if for all n, there is a mapping f„ : X„±1 . A spectrum X = {X n } is called a stable object ([Adams641) if for each n, X n is (n — 1)connected and the mapping fn : S X n X n+i is a (2n — *equivalence.

sx„

Spectra were introduced by [Lima591 and [Whitehead, GW621 to study Spanier-Whitehead duality and generalized homology theories. For our purposes, a full discussion of spectra and stable homotopy theory is not needed. The student of homotopy theory needs some exposure to these ideas. The books of [Adams74], [Switzer751, and [Ravene186, 92 1 are excellent introductions. If X = {X n } is a spectrum, then the cohomology and homotopy of X are defined as the limits: H' (X; k) = lim Hq+r (Xr ; k),

and

7rq (X) = ELT 7rg ±T (Xr)•

When the spectrum is a stable object, these limits are achieved at some finite stage (dependent on q). Furthermore, 7rq (X) is the gth stable homotopy group of XN, for some N = N (q) and Hq (X; k) = s' Hq+r (Xr ; k) for some r q. The construction of Adams resolutions may be applied without change in the case of a stable object. Proposition 9.29. Let X be a stable object. Then there is a spectral sequence with Et Ext' (H (X; 7,) converging strongly to ( ,)7rt _ s (X) under mild conditions. 1'

Though this appears to be a machine in search of a problem, in fact, these remarks apply broadly to the computation of cobordism groups as found implicitly in the work of [Thom541 and explicitly in the work of [Milnor60], [Liulevicius62], and others. Recall that two compact differentiable n-dimensional manifolds M1, M2 are (unoriented) cobordant if there is a compact differentiable (n 1)-dimensional manifold W with aW = M1 II M2 (disjoint union). Being cobordant is an equivalence relation and the set of equivalence classes of n-manifolds is denoted by 9I„. Disjoint union provides 91,, with the structure of an abelian group. The cartesian product provides the direct sum of the cobordism groups 91. with a ring structure.

9.4.

Lemma

409

Other geometric applications

9.30. '71L, is a vector space over F2.

PROOF: Given an n-dimensional manifold M, consider the (n+1)-dimensional manifold with boundary M x I. Since the boundary of M x I is M H M, we have that twice the cobordism class of M is zero in 0192 . Cobordism was introduced in 1895 by [Poincaré1895]. The homotopytheoretic study of cobordism was begun by [Pontrjagin55] who showed that the study of cobordism classes of framed manifolds is related to the study of By a theorem of [Whitney36] an n-dimensional manifold can be embedded in Rn+k for k large enough, and so a manifold is equipped with a normal bundle to the embedding. A manifold Mn c Rn±k is said to be framed if the normal bundle to the embedding is trivial. If we restrict our attention to framed manifolds only, then cobordism remains an equivalence relation and we denote the framed cobordism ring by 0,f[. A sufficiently small tubular neighborhood of M in Rn+k is homeomorphic to M x 1k• Projection off the second factor gives a mapping to Taking the one-point compactifications of Rn±k and Rk , we construct a mapp ing fm Sn+k sk by sending the complement of the tubular neighborhood of M in Sn± k to oc and going by the composite of the homeomorphism and second projection on the tubular neighborhood. This construction is well-defined up to homotopy and determines a class in 7rri+k(S k ). When we embed the manifold into Rn±k+ 1 via the canonical inclusion M c Rn+k c the construction yields Efm. Continuing in this way, we determine a class in 7rns . [Pontrjagin55] proved that all the choices made in this construction rei s well-defined. main within the homotopy class and so the mapping S2f*r Furthermore, it is easy to see that it is a homomorphism. An inverse mapping may be constructed by using some facts of differential topology. If g: Sn±k —> Sk represents a class in 7rns, then we can choose g to be smooth. Let p E S k be a regular value of g, that is, the differential dg: TS11 +k —>TS is of maximal rank for all x E g -1 ({p}). Regular points exist in abundance by the theorem of [Sard42]. The Implicit Function theorem implies that M (g) = g' (p) is an n-dimensional manifold whose normal bundle is trivial by comparing it with TS ) , a single vector space. To show that 1 f*r it remains to show that the construction from framed manifold to homotopy class provides the same element in 7r,s, , that is, fm (g) g. For complete details, see the classic book of [Stong68]. This construction was significantly generalized by [Thom541 in his thesis. The normal bundle vm over an n-dimensional manifold M embedded in Rn+k is classified by a homotopy class of a mapping F: M > BO(k). From the embedding, we can talk of a unit disk subbundle D <1 (UM) of the normal bundle, as well as its boundary, the unit sphere bundle S(v m ) in vm . The key to the generalization is the Thom space associated to the normal bundle, —

Th(vm) = D<1 (vm)/S(vm).

410

9.

The Adams Spectral Sequence

When M is compact, Th(vm) is homeomorphic to the one-point compactification of vm and so this construction may be carried out for any vector bundle over a compact space. When a vector bundle 77 is given by a Whitney sum, 7 =i e 6, then Th(71) Th(W A Th(6). By taking a limit over Grossman manifolds, there is a universal Thom space, 1110(k) = Th(-yk), associated to the universal dimension k vector bundle -yk over BO (k). The Thom space construction is functorial and so the classifying map provides a mapping Th(F) : Th(vm ) MO(k). The canonical inclusion 0(k) c 0(k + 1) corresponds to the addition of a trivial bundle to -yk and this provides a mapping fk : Al 0(k) = Th(-y k Ill) —> .1110(k ± 1). Thus the sequence MO = {MO(1), MO(2), MO(3), } )

constitutes a spectrum called the Thom spectrum. ([Rudyak98] has written an excellent book on the properties of such spectra.) The passage from cobordism groups 9I„, to the homotopy groups of the spectrum MO is made by taking a sufficiently small tubular neighborhood of M in Rn+ k that we denote by N. This space is homeomorphic to the open unit disk bundle D<1(M) with boundary S(vm). Suppose N C Rn±k C Rn±k U00 = S n+k ; we define a mapping f m: Sn+ k Th(um) as follows: Send N to D MO(k) represents a class in 7r,„ (MO) as a smooth mapping, then the inverse image of the zero section generically gives an n-dimensional manifold in S n±k and varying the representative remains in the cobordism class. The Whitney sum operation, as the mapping Wh: BO(r) x BO(s) —> BO(r s), provides a mapping of Thom spaces MO (r) A MO(s) 1110(r + s) that gives rise to a product on 7,.(M0). Since the Whitney sum of normal bundles represents the normal bundle to the product embedding, this product on homotopy groups corresponds to the product on cobordism groups.

Theorem 9.31 (IThom541). As rings, 01. —= 7. (MO).

9.4. Other geometric applications

411

It follows from Lemma 9.30 that (2 ) 7. (MO) = 7r. (MO) and so we can apply the mod 2 Adams spectral sequence and hope to compute directly the cobordism ring 01* . We turn next to the input to the spectral sequence.

Homology and cohomology of Thom spaces

In order to study the mod 2 cohomology of the spectrum MO as an algebra over the Steenrod algebra, we work on the individual Thom spaces in the spectrum. One of the main results of [Thom541 is the following computational toehold. Theorem 9.32 (the Thom isomorphism theorem). If —> B is an oriented k-dimensional vector bundle over a space B of the homotopy type of a finite complex, then, for any ring R, Hn± k (Th(); R) is isomorphic to Hn (B; R) for n > 0. Furthermore, there is a class Uk E Hk (Th(); R), corresponding in the isomorphism to 1 G H° (H; R), such that, for all n, Hn±k (Th(); R) Hri (B; R) Uk. SKETCH OF A PROOF:

homotopy

Recall that Th(C) = D <1 ()/S(). When B has the

type of a finite complex, we can write II* (Th();

= H* (D<1(0, S ; 110 •

We next apply the Leray-Serre spectral sequence for pairs (Exercise 5.6) to the fibrati on k s k - 1) (D <1 S ()) —> B. The E2-term is concentrated in the kffi row where we find E2n'k t-,.• fin (R ; B-k ( ek sk–i ; R)) pin ( B R ) . The orientation allows us to make this isomorphism globally. The theorem follows from convergence of the Leray-Serre spectral sequence and the cupproduct structure on the spectral sequence. (See [Milnor-Stasheff74] for a more geometric proof.) ( 6,

We apply this result to the universal le-bundle, over BO(k). Following the discussion in §6.3 we know that H* (B0(k); 2) is isomorphic to Fr) [wi , , wk] where the wi are the universal Stiefel-Whitney classes and deg wi = i. The wi may be defined from the symmetric functions on classes yi , • • • , yk of dimension one in H* ((B0(1))x k;F2)■ where OW x • • • x 0(1) 0(k) is the inclusion of the diagonal matrices with entries IL (0(1) The Thom isomorphism theorem gives H* (1110(k);72)

H* (B0(k);72)

Uk

2[W11 • • • 1Wk]

Uk.

The zero section provides a map BO (k) 11/10(k) that is compatible with the structure maps induced by the inclusions, j: 0(k) 0(k +1). It follow that

412

9. The Adams Spectral Sequence

there is a commutative diagram: s -k H* (M0(k);7 2 )

+ 472) (e) -1

(t*) -1

H* (B0(k + 1);72)

Bi*

H* (B0(k);72)

where the vertical maps are the inverses of the Thom isomorphism and the top horizontal map is induced by the spectrum map EMO(k) MO(k + 1). It follows that we can identify H* (M0;1 2 ) with lim<_, H* (B0(k);F2) = H* (B0;72) as a vector space. There is a coproduct structure on H* (BO; 72 ) that is induced by the Whitney product Wh: BO(r) x BO(s) BO(r + s) and H* (BO; F2) is a Hopf algebra with this coproduct. As an algebra H* (B0;72)

72[wi, w2, • • • , wk, • ..1.

The coproduct formula is given by the Whitney sum formula (Lemma 6.42). This yields a commutative product on the dual of H* (B0;72) and the Hopf algebra in this case is self-dual, that is, as algebras,

H.(BO: 72)

72[al, a2, • • • • ak, • • •

We turn to homology in order to avoid the noncommutative product on the Steenrod algebra. Mod 2 homology is endowed with the structure of a comodule over Alual , that is, there are homomorphisms gpx H* (X;E2) —> Alual ®

(X;72)

satisfying the dual axioms for the Steenrod algebra action. Thus the following diagram commutes for all spaces X: x

H* (X ;72)

Alual H* (X ;72)

) Alual H* (X; 7 2 )

Alual Alual

1

1011,x

B -*(x; 7 2 )

'ON

where qp: Ac2lual Aslual Adual is the coproduct on the dual of the Steenrod algebra. To study H* (M0;79), we notice that the Whitney sum induces a pairing MO(r) A MO(s) > MO(r + s) that commutes with the zero sections and the Whitney sum map on the classifying spaces. With some care in identifying generators (IStong68, Chapter VII), this induces a product on H* (MO;79) and gives an algebra isomorphism —

H* (MO; 72) = 1F2 [cti, a2, • • • , ak, • • • 1. —

9.4.

413

Other geometric applications

Recall the theorem of Milnor581 on the structure of Alual (Theorem 4.47): At the prime 2 we have dual

r=" F2[61,

• • • 6k, • • • ],

where deg 6i = 2 — 1. Consider the naive splitting: H * (MO; IF 2)

2[ai i = 23 — 1, j = 0, 1, .. • ] 01E2 [ak

k

23 — 1, k > 2].

This decomposition suggests that I1 (MO; 72) may be isomorphic to an exAdual 0 F2 [ak k tended comodule over Alual , that is, H*(M0;72) 2i — 1,k > 2]. An interpretation of the results of [Thom541 by [Liulevicius62] in the setting of Hopf algebras leads to a proof of this splitting.

Theorem 9.33. 11* (MO;F2) Alual ®72[ak k 2 3 —1,k > 2] as extended comodules over Alual . SKETCH OF PROOF: Recall that H* (B0(1); 72) (x i ), the divided power algebra on a generator of dimension one. The main ingredients in the proof are: (1) The fact that Bi * : H* (B0(1); F2) H* (B0; 72) is given by 7k (x1) ak . This follows from the dual representation of the universal S tiefelWhitney classes as symmetric polynomials in the one-dimensional classes in H*(B0(1)xk;7 2 ■) (2) The determination of the Alual -comodule structure on 11* (B0(1); F2) and hence, by virtue of the commutative diagram,

H* (B0 (1) ; 72)

Alual H*(B0( 1); F2)

Bi *

H*(B0;72)

loBi *

1Pso

• dual

11*(B0; 72),

the A 1 -comodule structure on H* (BO; 72) can be determined. Since the algebra structure on If (BO; 72) is compatible with the Aual -comodule structure, it suffices to check on generators. The comodule structure may be written 0 ® ai where o7 E (A2dual (4) There is another com'OBO (an) = mutative diagram that allows us to determine the Alm] -comodule structure on H* (MO; 1F 2): H* (BO; 72)

IP Bc,

,A (21"1 0 H*(BO;F2) liet*

t*1

H*(MO;F2)

Omo

)

Ac2lual (x) B-*(mo;F2).

414

9.

The Adams Spectral Sequence

Since the classes coming from MO(1) are identifiable with -yk(xi) — al, we get a map (B0(1); F2) —> H.(MO;F 2) given by -yk (s i ) ak_ . This gives the crucial step as we can compute '(bmo (ak) =

1+ Ei3O

if k = 2' — 1, i® ak decomposables if k 2. ® a1

(5) Finally, we can check that li)mo (a2- —1) = Ers=0 8ea2s-i. This follows from the representation of the dual classes wk as symmetric polynomials and the pairing of the Steenrod algebra and its dual (Proposition 4 of [Liulevicius621). A complete exposition of all these details may be found in [Schochet711 or from the cohomological point of view in [Stong68].

It follows from the theorem that H* (MO; 72 ) is a free module over A2 and so 7F* (MO) is computable immediately from ExtA, (H* (MO; F2), F2). But this is the dual to the generating module for the free module. Corollary 9.34 aThom541). 7r* (MO)

72[ak

k

2r — 1,k > 2] as

algebras.

Thom proved that there is a weak homotopy equivalence between MO and a product of Eilenberg-Mac Lane spectra Ek (w ) IKE/21' with the Is(w)1 given by the degrees of homogeneous polynomials s(co) in F2[ak k 2' 1. k > 2]. The role of characteristic classes in distinguishing nontrivial classes is crucial—one of the main theorems of [Thom541 is the sufficiency of the mod 2 characteristic numbers in classifying cobordism classes of unoriented manifolds. Thom's construction of the Thom spaces admits considerable generalization. In particular, we can define the oriented cobordism ring S2s* ° by admitting only oriented manifolds M1 and M2 and requiring that a cobordism W be orientable with boundary M1 II (—M2), where —M2 is the manifold M2 with the opposite orientation. The normal bundles of such manifolds have a lifting of their classifying map : M —> BO(n) to : M —> BSO(n). The universal n-dimensional vector bundle over BSO (n) has Thom space MS0 (n) and the same argument for unoriented cobordism can be made to prove that the spaces MS0(n) form a spectrum MS0 and 7F*(MS0) Qr. If one focuses on the lifting of the normal bundle to some classifying space BC —> BO(n), then a very general notion of cobordism is possible ([Stong68]). For almost complex manifolds, there is a lifting of the normal bundle to BU(n) B0(2n) and Chem classes and numbers distinguish cobordism classes. In a classic paper that introduced the application of the Adams spectral sequence to cobordism, [Milnor60] proved that 0.s° has torsion only at the prime 2, and that Se; is torsion-free. Furthermore, Milnor computed 7F* (MU) by determining the structure of H* (MU; 7,) for all primes p, and

9.5. Computations

415

then using the Adams spectral sequence. In fact, H* (MU; Il) is free over a quotient Hopf algebra of Ap and a change-of-rings argument (Theorem 9.12) allows the straightforward calculation of (p).74 (MU). The homological simplicity of complex cobordism led [Novikov671 in his study of a generalization of the Adams spectral sequence founded on MU. See the book of Mavene1861 for more details. Similar arguments were first carried out by [Ray721 and Mochman801 to obtain partial results for symplectic cobordism (Mochman961 gave a different approach to calculating 7. (MSp)). 9.5 Computations

Our point of departure is Corollary 9.23—there is a spectral sequence with E2-term isomorphic to the cohomology of the Steenrod algebra A p , and converging to (p)7r*s . Thus the problem of computing (p)7rs breaks into the problems of computing H*,* (A r), and then the differentials in the Adams spectral sequence. Finally, there is the problem of determining the extensions. We first construct a small part of a minimal resolution for A3, the mod 3 Steenrod algebra. The computation begins easily enough and you even get some of (3)7r, but it quickly gets complicated. We then consider the case of p = 2 more systematically. Following [Adams60], we are able to describe Hs'*(A2) in some detail for s < 3. Next we put these computations to work and find the first nontrivial differential in the spectral sequence. A corollary is the first case of the Hopf invariant one problem. We continue the hands-on computations with a discussion of Massey products and their relation to Toda brackets. Low-dimensional calculations

We begin by constructing the beginning of a minimal resolution, 6

do dl

0 <— 73

A3 <— P1 .— P2,

up to internal degree 9. For the most part, the discussion will be descriptive; the reader should construct a chart of everything that is happening. By Lemma 9.11 and Theorem 4.45, P1 has A3-module generators ao of degree 1 and hi of degree 4 3i = 2 (3 — 1) for i = 0, 1, 2 ... . These generators correspond to and P3' ; the homomorphism d0: P1 —> A3 is given by do (al)) = /3 and do (hi ) = P3 . In the kernel of do there is already Octo since 02 = 0. We put a generator do in P2 with di (do) = /3 a°. The next phenomenon to arise in the kernel of do that is not accounted for by do (ao ) occurs in degree 9. The diligent reader making a chart will find P2 0, OP2 , P1 11P1 and ,(1/31 P1 in A3 in degree 9. The Adem relations imply that only

9.

416

The Adams Spectral Sequence

two of these expressions are independent. In degree 9 in P1 there are three independent generators, P 2 a0 , /3P' h0 and Oho . The Adem relation plopl = oF•2 ± P20

implies that the element P2 a0 — P 1 ,3h0 — /3P l h0 is in the kernel of do. Let gi be in P2 in degree 9 with d1 (g1 ) = P 2 a0 — Pl i3ho — OPl ho • The chart should be getting a bit complicated by now. However, two simple patterns emerge: (1) Except for the element do in P3 with d2 (do) = 3ão, the first generator to appear in P3 is in degree greater than 11. This implies that we have computed Hat (A 3 ) for s < 2, t < 11; (2) The recurring Bockstein that arises at each stage behaves systematically and so, if we remove the chain of generators due to /3 2 = 0, the connectivity of this minimal resolution implies we have computed Hs , * (A3) for t< 11 and all S. Because the resolution is minimal, Ex-CA:7 ( 3, F3) = (T3 0.43 and so we have computed

Ps dual , )

(s, t) = (0, 0), (1, 1), (2, 2),

73, Exe;i: (73 T3) =

(3, 3), (1, 4), (2, 9),

elsewhere for s < 3 and t < 11.

{0 } ,

Since .E.,V is related to (3)7i5 s we display Exts' t with t —s running horizontally and s vertically. The differentials d, then lower t — s degree by 1 and raise s degree by r (that is, d, goes left one space and up r spaces in the (t—s, s)-plane). In the spectral sequence, we can display these data in the diagram: ,

3

ao

2

aci

1

ao

o

1

o

1

g1

ho

2

3

4

5

6

7

8

9

t -s

With this chart, the connectivity of the resolution, and the evident lack of differentials, we have computed the following stable homotopy groups.

9.5. Computations

Proposition 9.35.

Z,

if n = 0,

137-7

(3) 7rns =

417

{0 } ,

if n = 3 , 7, i fn = 1, 2, 4, 5, 6,8, 9.

The reader should compare this method with the method of killing homotopy groups in Chapter 6 (Corollary 6.27).

If the reader has been creating a chart to keep track of the minimal resolution, it should be clear that a systematic method of computation is desirable. We change to the prime 2 and study H* , * (A2) more systematically with the tool of choice—the change-of-rings spectral sequence (Theorem 9.12). Theorem 9.36. Let g denote the bigraded algebra XiXi±i =

Ç

= 7 2[X0,

xi, X21. • •

Xi2 Xi+2 X 3i+3 = XiXi242 =

with bideg xi = (1, 2 1 ). There are elements h i in H 1, * (A2) for i = 0, 1, 2, ... such that bideg h i = (1, 2') and a mapping ct: H"(,42), determined on generators by xi 1—, hi ; a is a well-defined mapping of bigraded algebras and a restricts to isomorphisms, H 1, *(,42) and g2 , - B-2 , .(A2 ), and to a monomorphism g3 , . H3, *(,A2). All relations among products of the generators hi are consequences of this mapping for Hs , ' (A2), s < 3.

PROOF (Mdams601): To compute H* , * (A2), consider the supporting cochain complex for this algebra, F. (Ad?al ) given by the cobar construction on Aual . Recall the coproduct for ,,4 1"1 = F2 [ .1, ion the generators is given by the formula of [Milnor581: V* =

l±k=i3

(6 = 1 ).

We proceed by a series of remarks: L Hi,* (A2) = 72{hi i = 0, 1, 2, .. , and deg hi = 2i } . This follows from Lemma 9.11. To determine Prim(4 l ) observe that i is primitive and so, because we are working mod 2, a' is also primitive for i > O. The formula for (p* shows that these are the only primitives. Let hi denote the class [a' ] in the cobar construction. II. If hi ho = 0, h8h2 + h = 0 and hoq = 0, then, for all i > 0, hi±ihi = 0, 1-qhi+9 + h41 = 0 and hi l4+2 = O. Because A"1 is commutative, we can apply Theorem 9.10: Sq ° exists in the Steenrod algebra that acts on H* , * ( 42) and the Cartan formula shows that Sq ° is multiplicative. Since Sq ° = Sq ° ([a]) = [a'+1] = h+1, repeated applications of Sq ° to the initial identities obtains the identities for all i > 0.

9.

418

The Adams Spectral

Sequence

III. hi ho = 0, 14122 +14 = 0, and hoq = O. Consider the formulas in the cobar construction: d* (

(1*([2

d*([3

I

a] + [a

)=

21+[62_ I 6_21+Ea2 I 6]) =

a -2] -F

[a a])

I , I6 I

']+ [

16].

=

These formulas imply that a : g H*,* (A 2 ) is well-defined in degrees s < 3. It remains to prove that a on g1 , * and on g2 , * gives isomorphisms and on g3 , * a monomorphism. Because we can describe ,,z1 1"1 so explicitly, we can find sub-Hopf algebras of ASlual of particularly simple form. Let Ba = F2 . . . -11 ]; then Br,' is a sub-Hopf algebra of 11S1"1 . Furthermore, we have the short exact sequence for each n, 0 —> —> —> —> 0 where A'n = I I Bry, [ -rt ], the Hopf algebra with dualizing, we obtain the extensions of Hopf algebras

primitive. By

0 —> An B a B a- 1 —> O.

Notice that (B ni )r (ASlual )r for r < 2n — 1. Thus H* ,t (B „) 1-1-*'t (A2) for t < 2n 1. We will use these extensions with the change-of-rings spectral sequence to compute H* , * (A9) in the desired range. To apply Theorem 9.12, we need a further remark. IV. An is central in B. Because we are not giving explicit descriptions of An and Ba, we consider the dual situation and ask: Does the following diagram commute? —

Bin ok

.14 0 .14 —•

14

Because (p* is multiplicative, it suffices to check the commutativity on the The explicit formula for the coproduct algebra generators, • • • , ' implies that the diagram commutes. By Theorem 9.12, for each n, there is a spectral sequence, converging to H' (Ba), with Er given by Hq (An) 0 HP (B_ 1 ). Because An, is (r - 1)connected, A2 = lim Ha, and so we get better and better approximations to H* (A2) with each H* (Ba). t't •

419

9.5. Computations

V. H* (A n ) [h, j = 0, 1, 2, ... ], where denotes the class [-„, ] in the cobar construction on A. This follows because A Il 2 []. With these data, we begin an induction. For n = 1, H* (BO

H* (Ai) `=" 7 2[hi i = 0, 1, 2, ... ]

where the hi corresponds to [a] of degree 2. For n = 2, g'q Hq (A 2 ) ® HP (BO. First examine d2 : E2" E22,0 ; d2 [2]) = 6] or d2(h2,0) = hi ho. When we apply Sq° , we obtain d2(h2,i) = hi+i hi . Thus d2 is monic on E20 ' 1 and, since H i (B2) comes from E oe an d Ey E2i3 O Hi (BO, we have shown H 1 (B2) L' (BO with the isomorphism coming from the projection B2 —> B1. Notice further that we have introduced the identities h 1 h, = 0 into H* (B2). Consider next d2 : E21 ' 1 E23 ' ° ; E21 ' 1 is isomorphic to H1 (A 2 )0 H1 (BO F2{h2,,i 0 hk}. Since d2 is a derivation, d2(h2,,i 0 hk) = hi+ihihk. This differential is a monomorphism except when the dimensions conspire to give a kernel. In particular,

Hi

d 2(h2,i ® hi+ 2 +

h2,i +1

0. 0 hi) = hi+ihihi+2 + hi±2hi+ihi =

Because no other differential is defined on E214 , these classes live to _Ee and determine elements in H2 (B2) that we denote by =

0 hi+2 h2,+1

hi }.

These classes lie in t degrees 3 • 2i + 2i+ 2 = 7 • 2, and so they are linearly independent. To finish our description of H2 (B9), we must determine EY. The differential d 2 : E2cr ' 2 E224 is given by d2(h2,ih2, 3 ) = h2, 3 g hi±lhi h2, ti hi±ihi, and so is nonzero except when i = j. Thus only the classes (h2,i) 2 survive to 4, 2 . To determine d3 : E3° ' 2 —> E33 '° , we consider (h2,0) 2 . In the cobar construction, this is the name for [ -r) -9 1 ; by III, this element is congruent modulo the filtration to K'2 21 + + [a6 -1,] which is carried by d* Thus d3((h2,0) 2 ) = h(2) h2 + hT. Repeated + V application of Sq ° gives

[a a a].

to

d3((h2, 9 ) 2 ) = 11,2 129+2 + h41 .

Therefore, EV = {0 } . Because the spectral sequence converges to H* (B2), we have the short exact sequence 0 —> EL° —> H2 (B2) —> —> 0.

9.

420

The Adams Spectral Sequence

We can describe EY as 2{hihi j i + 1} and Ee as 72{g2,i i = 0, 1, ... }. Recall the t degrees of all of these elements; hi hi c Eg.p+2' and - 2' • . We deduce that H2 (B2) is the direct sum of EY and EL'. g2,1 G Finally, we determine H3 (B2). In H3 (B1 ) we have introduced the relations hi+lhihk = 0 and filhi+2 + /4+1 = 0. We compute d2 : E2l ' 2 —> Again, d9 is a derivation, so on elements h9, i k2, 1 (S) hk we have d2(h2, i h2, 1 hk) = h2,i o hi+ihi hk + h2,i o hi+ihihk•

Unless i = j, d2 is nonzero, and so E31 ' 2 = 2{h2,ih2,i 4}. A subtlety enters here: d3(h2,h2, o hic ) = qhi+2hk + h4i hk follows from the derivation property of d3. It is possible for a relation to produce elements in the kernel of d3 if that relation is induced by d2. Consider d3 (h2,ih2,i

0 hi+3 h2,i+i h2,i+i 0 hi+ ) = qhi+2hi+3 h41hi+3

114

1142 hi+3.

The first and last terms are 0 since hi ±ihi = 0 and the middle vanishes as a pair. Since no other differential affects we have determined a flock of classes in E 2 , denoted by f2,i = {h 2,h2 , 2' = {[2

o hi+3 + h2,i+ih2,i+1 o hi+1 } 2'

2'+3

2i + 1 [2

2i+1

c2

2i+1

cl

]}*

Once again, the t degrees of the f2 ,, show that they are linearly independent. 11-2(A2) o H2(B1N) E22 ' 2 . The Next consider c12: E2°'3 B-3(A 2 ) formula d2(h2,ih24h2,k) = h2,A2,k0hi±lhi+h2,ih2,k0hi±lh1+h2,ih2,i0hk+ihk

is enough to show that d2 is a monomorphism; therefore, g:.? = {0}. To finish off H3 (B2), we consider d2: E22 ' 1 E24 ' 0 . A class in E22 ' 1 is a sum of classes of the form h2,i O hihk, and d2(h2,i O hihk) = hi+lhihihk. Many relations can be obtained by manipulating subscripts; however, most of these are generated by the image of d2 : E2 ' 2 E22 ' 1 and so are known. The exceptions are of the form hi+2hk + h2,+1 O hihk, which is seen to be g2,ihk. These classes are missed by d2 and so give permanent cycles. However, they are not linearly independent in the rest of H3 (B2). VI. g2,i hi+i = 17,42 hi in H3 (B2 ). To prove this, observe that in the cobar complex,

d*( e3 I e2 ] + [a I eie2]+ [e4 I [

=

Ea I a I ell+ V I e2 I

+ [e4

I el I al,

9.5. Computations

421

which represents g2,0111. Since Sq ° g2,i = g2,i + 1, this formula proves VI. If the reader has kept track of the new elements in H3 (B2 ), then, because the internal degrees of these elements all differ, we have shown that H3 (B2) contains 72 {hi k hk i j +1, j k + 1} modulo h?hi +2 = 124,, as well as 72{f2,i I i = 0, 1, ... } and 72 {g2,ihi g2,ihi±i= hi3±2hi }. We next proceed in our induction to H* (B3). E22,0 For n = 3, _4' Hq (A 3 ) ® HP (B9) . First examine d2: E2" d2(h3,0 = {d'([6])} =

6_]+

[

6]} = {h2,10ho+h2,o0h2} - g2,0.

When we apply Sq°, we obtain d2(h3,i) = g. Thus d2 is monic on E2" and therefore, EW = {0}. This shows 111 (B3) H 1 (B2) (B ) with the isomorphisms induced by the projections B3 -> B2 Bt Consider next d2 : E21 ' 1 -> E'0 . By the derivation property of d2, we have d2(h3,i 111) = g2,ih1 . If j = i + 1, this still makes sense and we get d2(h3,i hi+t) = = h4 2 hi, which introduces a new relation (B3). Looking at sums in E21 ' 1 among the 3-fold products of elements in we consider classes h3,i hi+3 + h3 ,i+ , 0 hi . Apply the differential to get d2(h3,i hi + 3 + h3,i + 1 0

hi)

- g2,ihi+3 + g2,i+ihi = {h 2 , i hi+ 2 + h2,i+1 = 0,

+ {h 2 , +1 hi + 3 + h2,i+2 0

since h1, h1 = 0.

This determines classes g3,i = {123,i ® hi+3 h3,i+1 hi} in _E,2 and so in H2 (B3). The classes g3,i lie in t degrees 2i(23 2i + 3 = 15 • 2i and so are linearly independent. To finish the description of H2 (B3), consider d2: E2cr ' 2 -> E22 ' 1 : d2 (h3 , i h3 ,i ) = h3,1 0 g2,i + h 3 , O g2,j,

which is nonzero unless i = j. Next consider d3: E3° ' 2 -> E:' ° on the remaining classes (h3,i) 2 . In the cobar construction for B3, [6 6] is identified modulo the filtration with [6 31+ [23 I i]+[ I 2]+[ I '16]+ I 66]. The differential on this sum is r V} which repre+ [62., sents {h 0 h3 h 2, , h 1 } = f2 , 0 . By applying Sq ° , we have shown d3((h3A = f2,i•

Thus H2 (B3 ) is the direct sum, F2 {hihj

± 1 } ED 72{g3,i i = 0, 1, ... } .

Next, let us consider H3 (B3 ). Notice that in H3 (B3 ) the relations hi +jhihk = 0,

hihk+thk = 0, h?hi+2 + h4 1 = 0

and h42 hi = 0

9.

422

The Adams Spectral Sequence

hold for 3-fold products of generators of H' (B3). In order to show that these are the only relations, we continue the induction and we determine more of H3 (B3).

E93 ' 1 . The formula,

Consider d2:

d2(h3,ih3,1 g hk) = h3,1

g2,ihk 0 g2,1hk,

shows d2 is nonzero unless i = j, so we have E31 ' 2 = 7 2{h3,ih3,i Furthermore, d3((h3,i) 2 hk) = f2,ihk. This produces a kernel, however, when we consider the representative expressions for the f), i : d3 (h3 , i h3 , i

hi+ 4 + h3,i+ih3, i +

)

= f2,i hi+4 + f2,i+1 hi+1

= {h2,ih2,i ® hi +3 +

® hi+i}hi+4

+ {h2,i + ih2,i+ i ® hi + 4 + h2,i+ 2h2,i + 2 ® hi + 2}hi ± i = 0.

We denote the classes {h3 ,i h3 ,i hi+4 + h3,i+i 1i3, i+i hi±i } by f3 , i in The t degrees of the f3,i are 15 • 2i±' and so they are linearly independent. Next, we leave it to the reader to check that d2 and d3 map E2° '3 and E3° ' 3 in such a way as to leave Ey = {0 } . In E22 ' 1 , we find the classes g3,ih1 left over after clearing the image of d2 : E2° ' 2 E224 and the classes mapped nontrivially by d2: E22 ' 1 —> Er. If we check t degrees, however, it is possible for more than one g3, i h1 to inhabit the same degree. The classes that require comparison are g3,ihi-1 and .g3,i—ihi+3. Writing them out, we see that g3,i hi _ i = Ih3,i 0 hi+3hi_1 + h3,i+1 0 hihi_i = g3,i_ihi+3 = {h3,i-1 hi+2hi+3 + h3,i 0

h3hi _ 1 1, = {h3,i hi-ihi+31,

and so we must introduce the relation g3,i hi _ i = g3,i_ihi+3. Some amusing number theory can be employed to show that the other classes occur in differing t degrees and so we find classes g3,i hi, for i j + 1, all linearly independent. Thus H3 (B3 ) is seen to contain 72 {h i h1 hk i j + 1, j k + 1 } modulo the relations hihi+2 = /4+1 and 1 42hi = 0, 7 203,i j = 0, 1, ... 1 and 72{93,ihi i j + 11.

We can now summarize the inductive step in a series of formulas that extend the pattern above: (1) There is a spectral sequence, converging to H* (13) with Ef' q Hq (A n ) e HP (Bri _i)• (2) H 1 (B n _ 1 )

72{h i = —2

= 0, 1 , ... } j 1 } 0 7 2 {grt-1,i 1-P {hi hj i H3 (B_ 1 ) contains 72 {h i hj hk} modulo the relations hi + ihihk = 0,

9.5. Computations 3 h?hi+2 hi+1 = i = 0, 1,

423

0 and h2i + 2 h- — 0 plus the direct j 1}. i

sum F2Llen-1,i

7 2{9n,ihi

}

(3) d2(hn,i) = gn-1,i, d3((hn,i) 2 ) = fn-1,i•

(4) There are new classes in

= {hn,.

and so in H2 (Ba) given by

hi+n +

hi — gn _ i , i hn±i . The These classes are a result of the relation bin relation g, +i h = gn,ihi+n+1is seen to hold for the new gn,i • (5) There are new classes in E 2 and so in H3 (Ba) given by {h

hi±ti

h n ,i+ 1 h n ,i+1

hi+1 }.

These classes are a result of the relation hk + hk = 0, and the expressions for The reader can check that the t degrees of the new classes grows to infinity n goes to infinity. What is left in the limit, H* (.49), is the set of classes {h j = 0, 1, . } and the two and three fold products, subject to the relations

as

we have derived. This proves Theorem 9.36. The chart on the next page summarizes the data for H* (A2) given in Theorem 9.36 and for a small range of t — s. We follow the convention of writing Hs ,t (A2) on the lattice point (t — s, s). We also use the convention (due to 1Tangora661) of joining two classes together by a vertical line if one is the product of the other with ho, and by a line of slope 1 if one is the product of the other with hi • The diligent reader can write out the first few stages of a minimal resolution to see that nothing occurs above filtration degree 3 for 0 < t — 3 < 5, and the tower of g continues to infinity (this also follows from the discussion at the end of §9.3 and the fact that ( 2 )7r =

3 2 1

0

ho h1

h3

h2

h4

1 0

1

2

3

4

t

5 9

6

7

8 6 10 11 12 13

14

15

9.

424

The Adams Spectral

Sequence

Corollary 9.37. E/2E,

if n = 1, 2,

E/SE,if n = 3, if n = 4, 5.

{0},

PROOF: From the diagram we must first dispense with the possibility that d (hi) = hP for

some r. We use the fact that dr is a derivation and compute 0 = dr (h i ho) = hodr (h i ) =

0,

a contradiction. Therefore, dr (hi) = 0 for all r and h1 is a permanent cycle. As for the E/EC ( 2)4, the discussion at the end of §9.3 shows that the relations h2 0, h0 h 2 0 and 128122 0 describe an element in (2) 71 with 4 times that element nonzero. This forces the composition series for (2)74 to be 0 c Z/2E c /7 /4E c 7Z/87, = (2 ).

In order to do further calculation of (2)7T ' , more of H* , * (A2) is necessary than computed in Theorem 9.36. In fact, not all of H3, * (A2) is given by 3-fold products, as we see later. A more powerful technique for computing H* , * (A2) is given in §9.6. In the meantime, we obtain some geometric consequences of our computa ti on s .

The first nontrivial differentials The problem of the existence of elements in 7r ' with Hopf invariant one has been reduced to the study of the Steenrod operation Sq 2 ' and whether it acts nontrivially on a two-cell complex. Because Sq 2' is dual to a in Adual , the Adams spectral sequence further reduces the question to the survival of hi in to E 2'.

If hi survives to E 2 , it detects a class in (2) 7T The stable homotopy ring qr is graded commutative and so the square of the class detected by hi has order 2 because [hi] o [hi ] = o [hi ] and so 2[N] o [hi ] = O. This implies that the class ho fq that represents 2[hi ] o [hi ], which is nonzero if i > 3, cannot survive to Eoo by the identification of products with ho in Ext with the doubling map on 7„8 . The kind reader will forgive the following algebraic deus '

ex machina: Fact. Ext A2 s ' s+13 (IF 2,F 2) = {0 } for all

S.

With our present techniques, the proof of this is a good day's work constructing a minimal resolution. More elegant and streamlined techniques will be presented later. Notice that this implies (243 = O.

9.5. Computations

425

Proposition 9.38 ([Toda551, [Adams581). There is no element of Hopf invariant one in 731 (.9 16 ) = 71 5 and so there is no division algebra structure on R16 .

PROOF: We consider ho q in E23 ' 17 . Because E2s 's+13 = {0} for all s, no differential on h0h3 has a nonzero image and h0h3 is an infinite cycle. Since it cannot survive to E„, it must be a boundary. There is only one possible nonzero differential, that is, d2 (h4) = h0h3. Hence h4 does not survive to E„. We remark on the extraordinary blend of topology and algebra in this proof. The graded commutativity of and an Ext computation together imply a deep result. The next question to consider is whether this technique propagates through the spectral sequence to settle completely the Hopf invariant one problem? A quick glance forward to the charts in §9.6 shows that we cannot naively proceed even to h5 (in particular, H'29 ( A2 ) {0} for some s). Something more is needed. Our goal for the rest of this section is to outline how to prove: Theorem 9.39. d2(hi) = hoh?_,, for i >

In a celebrated paper, [Adams60] gave the first proof of this theorem based on a generalization of the factorization of Steenrod operations of [Adem521. The equation d2 (h4) = h0h3 is equivalent to a nontrivial factorization of Sq l6 into products of primary and secondary cohomology operations. Therefore, Sq 16 cannot act nontrivially on S' 6 U n e32 for any o in -,731 (S 16 ). The factorization is based on secondary operations, (1)i,i that arise from the Adem relation Sq 2' Sq 23 = 8 sq 2 < j _ 1). [Adams60] showed

E o<.9 <; b

that a decomposition, Sq 2 " = E ai with G A2 and modulo some indeterminacy, holds for all n > 4. This settles the Hopf invariant one problem. We note that Mott-Milnor581 and Mervaire581 had also settled the division algebras question shortly after Adams by using K-theory techniques. [Maunder63, 64 1 developed the notion of higher order cohomology operations and related them generally to differentials in the Adams spectral sequence. The connections between these operations, Theorem 9.2 and the operations in §9.1 were clarified by [Maunder63, 64 1 . Another way to show that d2 (hi ) — h0 h_, is to consider the Steenrod algebra that acts on Ext:Z (T2,72) and the fact that this action intertwines with the differentials in the spectral sequence. A priori, there seems to be little connection as this action is a formal feature of the cohomology of a Hopf algebra. The missing geometric link was forged by [Kahn70] using the geometric construction called the quadratic construction on a pointed space, (X, oco): First form the space Sn o (X >< X), where UoV=Ux V/ * x V, and then take the quotient F(X) = sn o (X x x)/( 6 , x, x') (-9,4 x). "

9.

426

The Adams Spectral Sequence

The inclusion of Sn

Sn+ 1 as equator determines a natural transformation and the quadratic construction is the direct limit of the natural transformations: Fn (X) —> Fn+ 1 (X)

1-"(x) = nurn rn(x). [Kahn70] showed that this construction carries the chain homotopy that gives rise to the Steenrod operations on H* , * (A2) and so these operations can be related to the differentials. The theorem that applies to the question at hand is given in the formula of [Milgram72]: d2 (Sqi (a)) = hoSe l (a),

which holds if a Ext .st (72, 72) and i t (mod 2). Since hi is in 1 2' EXt;12 (F2, F2) and 2i 0 (mod 2), d2(Sq ° hi) = hoSql (hi), that is, d2 (h 1 ) Nit?, which holds for i > 1 by Theorem 9.10. This technique has been em- = ployed with great success to determine many of the known differentials in the Adams spectral sequence ([1(ahn701, [Milgram721, and [Bruner841). in §9.6 we return to the question of differentials in the Adams spectral sequence and discuss some other methods to determine them. Massey products

Before we leave the computations that can be done by hand, we fill in more of our chart by considering the analogue of Massey products for a bigraded differential algebra. Ordinary Massey products and their higher order analogues are discussed in §8.2. Ext r" (k , k) = H* , * (F) is computed from a differential bigraded algebra, (B", d) = (F. (Alual ) , d*), the cobar construction, with its differential of bidegree (1, 0). In such a bigraded algebra, suppose [u] is a class in Hs , t(B* , * , d), [v] in Hs' , t' (B* , * , d) and [w] in II"'" (B* , * , d), and furthermore, [y] [y] = 0 = [y] [w]. Then we can define the Massey triple product ([u] , [v], [to]) c Hs+ +,"-1,t+e +t"(B" , d)

by taking elements a in B 8+8' and b in 138 ' +.9 " +t " such that da = uv and db = vu', where u E [u], y E [y], and W G [W]. As in §8.2, ([u],[v],[w]) = {[aw ± ub]

all possible choices of a, b, u, y, w},

where we denote the homology class of an element t by [t]. The indeterminacy for ([u], [y], [w]) is given by [u]H"s"',t'-kt" H8±S'-1,t+e r wl . The higher order analogues of the Massey product can be defined as in §8.2 with the extra index kept in tow.

9.5. Computations

427

Theorem 9.36 presents many trivial products that can give rise to triple products in H* , * (A 2 ). We next record some nontrivial Massey products. In particular, we compute in detail that co = (g, ho, h 1 ) and //lc° are nontrivial. Other computations are left as exercises or given in the references. The identification of Massey products in Ext also follows by special spectral sequence arguments first given by [Ivanovskil'64] and by [May64]. We begin with an exercise for the reader. These relations were identified by [Adams60] and follow from the formulas in the proof of Theorem 9.36: (hi+1, hi, hi+i) =

=

hi+i, hi) = g2,i•

These dispose of the most obvious choices for Massey products. The next relations to try are hYto = 0 and hol-ti = O. In the cobar construction, these products vanish because the following formulas hold:

Eet I et I el] = d*([6 [6_

I

+

I

+

I

+ [e4 I di)

= d* ([6 +

Thus, in H3,11 (A2), (g,ho ,h 1 ) contains the class le3la[d

I 6 I e?]+[et de2 e?]+[d

e?]+[e I et I 6+all.

In fact, this class is the unique representative for (121 ho , h l ) since the indeterminacy of (4, h 0 , h,) is g Ext 1 ' 3 EXt 2 '9 = {0 } . To show (h. ,ho, hl) is nonzero, it suffices to show that it is nonzero in H3 (B4) since 11 < 24 -1. We first identify it in H3 (B3). Observe that the class can be identified with [6 modulo the filtration in the cobar construction for B3 and so it names the class h3,0 hi h i in = H1 (A3 ) e H, (B 2 ) in the spectral sequence converging to H* (B3).

h,

a a]

d2(h3,0 hihi) = g2,0hihi = qhohi — 0

and so h3,0 0 ittiti gives a class in 4 1 that persists to El i- . Checking the degrees of the other nonzero classes in H3 (B3), we find 03,0 0 hi hi 1, of t degree 11, is not accounted for by classes hihihk, f3,i or g3,ih1. To see that {h3,0 (5') ht hi } determines a nonzero class in H3 (B4), we consider the next spectral sequence and d2:

(A4) ® H i (B3 ) = E

= H3 (B3 ).

Since E21 " is 15-connected in this case, (4, ho, Ill) lives to H3 (B4) and hence to a nonzero class in H3 " (A2). A similar dimension counting argument can be given for the class , ho , hi)hi = {h3,0 hihihi}

9.

428

The Adams Spectral Sequence

Thus we have identified two new elements in Ext3 ' 11 and EXt 4 ' 13 , denoted by co and h1 c0 by [May64]. The cobar construction is a very large complex to use for computing Massey products. avanovskiT641 and [May64] worked in more manageable complexes from which to compute H* ,* (A2). We summarize some of their computations in low degrees. We picture only the Massey product elements.

11 10

2 7 8

9

10

11

12 13 14 15

t

16

17

18

19

s

Theorem 9.40. For t — s < 19, the following Massey products are nonzero and their products with ho and hl are given in the chart. (The operators Pl and P2 are periodicity operators that will be defined in §9.6.) (1) (2) (3) (4) (5) (6)

(7) (8) (9) (10)

ho, h l ) in Ext3 ' 11 , = (h1, h, h3) in Ext 5 ' 14 , P 1 h2 = h8, in Ext 5 ' 16 , do = (ho , 14 ho) in Ext4)-8 , co, h2, ) in Ext 4 ' 21 , eo = h3, ho modulo 1)3h2h4 in Ext 4 ' 22 , fo = ci = (h2, h1, N) in Ext3 ' 22 , Pic° = h3) in Ext 7' 23 , P2 h1 = h8, = ((h3, h61 , hzol , h3) in Ext9 ' 26 , P2 h2 = hg, h4) = ((h3, h, h9), h h3 ) in Ext9 ' 26 . CO

=

,

For a more extensive and complete description of Massey products in the cohomology of the Steenrod algebra, the reader can consult the paper of [Tangora941.

9.5.

429

Computations

If the reader adds this chart to the results of Theorem 9.36, then the description of Ext s't for t — s < 19 is almost complete. The missing data are given by a theorem of [Novikov591 that adds the relations 14N = qh2h4 0 and hiPt„

0 for k < 27n-1

[Novikov591 showed that hr hrri+, = 0 for m > 2. [Maunder651 showed the nontriviality of the products 1 oi h 71 + 1 for 1 < 2m by comparing the Adams spectral sequence converging to (2)7r: : with the Adams spectral sequence converging to (9)7r*S (BUt2q, , cO, the stable homotopy groups of the 2q-coconnected cover of BU. With these facts, the description in these limited degrees is complete. We discuss some of the differentials in §9.6. Massey products are a formal consequence of the structure of a differential graded algebra. In cohomology they capture higher order linking phenomena. We can ask if there is a topological interpretation of the Massey products in the Adams spectral sequence. The product structure on stable homotopy groups of spheres is identified with the composition product that shows up as the Yoneda product on the E2-term. We need the notion of secondary products for the composition product, introduced by [Toda591. Definition 9.41. Let -y E [X ,Y], 13 E [Y, Z] and a [Z, W]. Suppose 00-y and co 13 are null-homo topic in [X ,W]. Lei c, b, a be mappings representing 7, 13 and a, respectively. There are extensions of boc and aob to C X —> Z and CY —> W, which we denote by B and C, respectively. Write S X = CX U C — X and consider the mapping S X —> W given on CX as a o B and on CX as C o c. The set of all such mappings is denoted 0,1) c [SX, W. ], called the Toda bracket of a, 3 and -y. It has indeterminacy given by the subset a#[SW,Y] + (S-y)#[S X, Z].

The definition, like the definition for Massey products, can be generalized to n-fold Toda brackets and matric Toda brackets. Furthermore, if we apply the definition to representatives of mappings in 7t.s, by careful suspension, we can define Toda brackets of stable maps for which the Toth brackets represent cosets in 7t.s. [Toda62] gave extensive computations of 7,,±k(Sn) for k < 19, using this secondary bracket product to determine and name many elements. [Cohen, J68 1 proved that all of 7s can be represented by higher order Toda brackets applied to integer multiples of the classes z77 c v a E 7r7 and a i E 74p _ 3 for each odd prime p. Massey products of all orders may be defined in Ext E2. Their relationship to differentials can lead to a connection with Toda brackets. [Moss70] provided a description in some cases. Broadly stated, he proved that Massey products of permanent cycles, under certain conditions, converge to Toda brackets in 7r:! (in fact, the main result of [Moss70] applies more generally to the composition product o: {Y, W } . x {X, Y},, —› {X, W } ,0.

430

9.

The Adams Spectral Sequence

Theorem 9.42 ([Moss70]). Let (Er't ,dr) denote the Er-term of the Adams spectral sequence converging to (p) 7t' .

(1) Suppose a E Ers, t, b Then

E't', and c Ers" ,t" satisfy ab = 0 andbc =0.

dr (a,b,e) c (dro, a ,

b 0 c (-1)idrb b' (-1)i±i' drc '

the matric Massey product, where i = t — s, (2) If adrb = 0 and bdrc = 0, then dr (a, b, c) C — (d ra,b, c) — (-1)'

= t' — s'.

drb, — (-1)

drc).

(3) If a, b and c are permanent cycles representing a in , in 7rP,' and -y in 7rf,, with a o = 0 and 3 o -y = 0, then, under certain technical assumptions on the filtrations of a, b and c, the Massey product (1,, b, c) contains a permanent cycle that is realized by an element of the Toda bracket (Tt, -y).

The simplest example of this is given by the element co in H3 ' 11 (A2). Looking ahead to the more complete table, co is a permanent cycle and hi, ho and h1 satisfy the unspoken filtration conditions of the theorem. Thus ho, h1) represents a Toda bracket that is given in Toda's notation as (v2 , 2i, n). 9.6 Further structure

The element-by-element arguments of §9.5 led to many useful results; in this section we take a different point of view and discuss the spectral sequence for (p )7r, in more global terms. We begin with two deep theorems of [Adams66]. The first determines conditions on s and t for which Extsi, 2, 72) = {0}. The second reveals portions of Exts.lit, 2, T2) that are isomorphic via periodicity operators; this periodic phenomenon determines infinitely many nontrivial values of the E9 -term of the spectral sequence. We then return to the computation of H* , * (A p) to exploit the fact that the Steenrod algebra is a graded Hopf algebra. The method of computation is called the May spectral sequence and was introduced by [May64]. Tables for H* , * (A2) and H* , * (A3) in a range are given. We then discuss some of the techniques to determine differentials in the Adams spectral sequence. We close the chapter with some remarks on further developments involving the Adams spectral sequence. (

(

The vanishing line

The main theorem of this section gives a description of the cohomology of the Steenrod algebra in the large. The proof is due to [Adams61, 66] for p = 2, and to [Liulevicius63] for p, an odd prime.

431

9.6. Further structure

Theorem 9.43. Let U(n) denote the function on natural numbers given by U(4s) = 12s —1, U (4s +1) =12s +2, U(4s + 2) = 128 + 4, U(4s + 3) = 12s + 6. Then Ext )s tt, (72,72) = {0}, for 0 <s
The vanishing condition in terms of n = t — s, the stem dimension, is given by 0 < t — 3 < U(s) — 3. By definition, Er (3) — 3 < 23 and so the condition n < U(s) — s is satisfied when s > n/2. In other words, above the line of slope 1/2 in the (t — s, s) plane, Exts;it2 (F2, F2) = {0 } . This line is called the vanishing line. Recall that the exponent of a group G is the least natural number, m, such that all mth powers of elements in G are zero. The vanishing line of Theorem 9.43 puts an upper bound on the length of a composition series for the stable homotopy groups of spheres. Corollary 9.44. For n > 1, the exponent of ( 2)r is less than2f(n) where f (n) is the minimum of fsln < U(s) — s — 11. The exponent of ()7r is less than p9(n) where g(n) is the minimum of 1.31 n < (2p — 1)s — s — 31.

We give a proof of Theorem 9.43 in the case p = 2. We proceed by a series of lemmas to prove a slightly weaker result that is strengthened later by the periodicity isomorphisms. Consider another numerical function, T(n), given by T(4s) = 12s, T(48+1) = 128+2, T(48+2) = 128+4, T(48+3) = 128+7.

Let A(r) denote the subalgebra of A2 generated by {Sal „ q 2S'

,

}.

Lemma 9.45. Extsjit, (A (0), 1F2 ) = {0} when s < 4 and 0 <s
PROOF: Observe first that the suspension isomorphism on graded modules over a graded algebra satisfies the relation for t > r: HomPsr , N) = rMod(srM, stN) rMod(M, N) = Homtr—r (M, N). It follows that Extrs't (sr M,

Extrs't '(M,N).

Following [Adams661, we consider the extension over O

A2:

-> SF2 -> A(0) -> If2 -> O.

This extension determines a class in Ext 14,21 (8F2, F2) Ext .111,°, (72, F2), which is given by ho and corresponds to Sq l . The short exact sequence leads to a

432

9. The Adams Spectral Sequence

long exact sequence of Ext groups, here abbreviated as Ext7;tt, (872, F2) = Ext s' t (s72), Ext I s 2 (A(0), 72) = Ext s 't (A (0)): s 2 (72, 72) = H8 't , and Ext / Exts 1,t(s72) L -

-- 8,t

Ext s't (A(0))

Ext s't (s72)

L H8,t

The reader can consult the charts in §9.5 to prove the lemma in the bidegrecs stated (there are 11 cases). Whenever the boundary homomorphism a is nonzero, it is given by multiplication by ho, and the result follows. We next restrict our attention to a particular class of A2-modules. An A9-module L is an A(r)-module for any r because A(r) is a subalgebra of A2. Therefore we can speak of L being a free A(r)-module. The relation Sq l,Se = 0 provides a neat criterion for a module to be free over A(0): Take Sq l : L L as a differential and compute the homology H(L,Sq 1 ); an A2 module L is free over A(0) if and only if H(L,Sq 1 ) = {0 } . This follows because Sq l x = 0 if and only if x = y. (For generalizations of this idea see the work of [Adams-Margolis711, [Margolis831, and [Palmieri92].) Lemma 9.46. Suppose L is an A2-module, free over A(0), and L is (n — 1)connected (that is, Lt = {0} for t < n). Then Exts/2 (L,72) = {0} for s < 4 and° <8 m. Notice that L(n) = L, that L(m) is an A2-submodule of L and that L(m)IL(m +1) is a free A(0)-module on basis elements of degree m. Therefore, we can write

L(m)IL(m + 1)

smA(0).

We proceed by induction on m. Consider the short exact sequence 0 —> L(m)I L(m +1) —> L I L(m +1) —> L I L(m) —> O.

Lemma 9.45 applies, with a dimension shift, to have

L(m)/L(m

+ 1) and so we

Ext ; s 2 (L(m)I L(rn +1),IF2 ) = {0}

for s < 4, 0 < s Ext!_it2 (L I L(m +1),72) Extsf2 (L(m)IL(rn + 472)

provides the inductive step. Finally, for a given s and t one can find in large enough that Exe.Z(L,72) Ext:T2 (LI L(m),72). This proves the lemma. El

9.6. Further structure

433

Lemma 9.47. Suppose L is an A2-module, free over A(0), and (n — 1)connected. Then, for 0 < s
PROOF: Suppose we are given a short exact sequence of A2-modules 0 —> Li —> L2 —> L3 —> 0,

and suppose two of the three modules is A(0)-free. When we apply the functor H ( ,Sq l ) to the short exact sequence we get a long exact sequence on homology. It follows immediately that the third module is A(0) -free since its Sa l -homology must vanish. We use this observation in what follows. Lemma 9.46 gives us the lemma for s < 4, that is, for k = 0 and s = 4k +i, for i = 1, . , 4. Suppose the lemma holds for all modules and for values of

less than 4k + 5. Let 0

L

do

Co <— C1

di

d2

d3

C2 <— C3 <— C4

be an A2-free resolution of L. Since the Ci are A2-free, they are A(0)-free. It follows that the modules ker di are A(0)-free, for i = 0 to 4. Let M = ker d9 = im d3. For a minimal resolution, Lemma 9.46 implies that M is (n + 11)-connected. Therefore Ext

(M,IE2) = {0},

for s <4k + 4, 0 < s
and so the lemma holds for s less than 4(k +1) + 5. Corollary 9.48. Ext / s 2 (72, F2) = {0} for0 < s
PROOF: Consider the short exact sequence 0 —> I(A2)1 I (A2) • Sq l

A2/1 (A2) • Sq l

2

—>

0.

The minimal resolution for A9//(A2) • Sq l is particularly simple and implies that, for 8 > 0, t s, EXV:4 , t2 (A2 I (A2) • S'q i ,7 2) = {0 . }

It follows from the long exact sequence associated to the extension that Exe;t: (F2. 72) Ext A 8 -2 " (1- (A2)//(A2) • Sq l ,T2)

434

9.

The Adams Spectral Sequence

for t > s > O. The module /(A2)//(A2) • Sq l is 1-connected and free over A(0)—this follows from the formula Sq l Sq l- — Se when / = (i 1 ,... , ir ) (0), and j1 is even, while Sq l Sq l = 0 when j1 is odd. The corollary follows from Lemma 9.47 and the appropriate expression for V (n). 111 Notice that V (n) = U(n) for n 0 (mod 4); to prove Theorem 9.43 we only need to settle one further case. We do this next. The proof for odd primes is similar and the appropriate homological algebra was developed by [Liulevicius62]. Periodicity The use of subalgebras of A2 in proving structure theorems about Ext was very fruitful in §9.5. We continue to study the family of subalgebras A(n) = Sq',Sq 2 ,...,Sq 2a , the subalgebra of A2 generated by the indecomposables Sq l to Sq 2 ' . The first such subalgebra, A(0) is isomorphic to an exterior algebra on a single generator of degree one. We can picture A(1) as in the diagram, where 1 is the bottom element, and each circle is a basis element. The straight line connections are given by multiplying (on the left) by Sq l , and all curved line connections are given by multiplying by ,S'q2 .

We construct a minimal resolution of A(0) as an A(1)-module, from the following short exact sequence: di d2 0 <— A(0) <— A(1) <— A(1)a e A(1)b d3 A(1)u e A(1)v A(1)t

si2A(0) <_ 0,

where di (a) = Sq 2 , di (b) = Sq2 Sq l , d3(u) = Sq2 a Sq l b, d3(v) = Sq2 b, and d4 (t) = Sq 2 Sq l u+ Sq2 y (The reader is encouraged to make a chart of this resolution using pictures of A(1)). We can extend this sequence to a complete minimal resolution of A(0) over A(1) that is periodic of order 4 and degree 12. This implies the following result.

9.6. Further structure 1) (A(0), 72) Ext A 8-43

Lemma 9.49. Ext

435

t+12 (

A(U) 7 2),

for s > O.

The quibble at s = 0 has to do with the units in A(0) and A(1). An immediate corollary of the lemma is the isomorphism for A(1)-modules, free over A(0): Ext8't

Ext s+4,t+12 (L,72).

(Liu

-

A(1)

The proof follows the same lines as the proof of Lemma 9.46. Ext groups have an interpretation in terms of equivalence classes of finite exact sequences; the piece of a minimal resolution we constructed gives a class in Ext4A(1) ' 12 (A(0), A(0)) and the isomorphism in Lemma 9.49 can be obtained by splicing (Yoneda multiplication) with this class. There is also the composition pairing that determines a right action: s+07;,t+t' (L, 72 ) . Exts4_ ) (L,72) 0 Hs' 't ' (A(1)) —> ExtA

Our first goal is to describe the isomorphism between Ext J s (i) (L, 72) and s+4,t+12 (L, F2, Ext A(t) 11' )

as multiplication on the right with a class in H4 ' 12 (A(1)). From this viewpoint we will see how to generalize.

Lemma 9.50. Ext4'12 Ext4'12 A( 1) (1`2,7,>) A(1) (A(0),7 2 ) 7 2 . Furthermore, right multiplication by the unique class ful E Ext 4 12 (F2, F2) gives the isomorphism Ext

s+(i4t+ 12 (A(0), 7 2). i) (A(0), 7 2 ) ExtA

PROOF: Consider the short exact sequence of A(1)-modules: 0 —> s72 —> A(0) —> 7 2 —> O.

This induces a long exact sequence of Ext groups: H 3,1104))

„, J_))

Ext 4Z2) (A(0), I2)

H4,11(A(1))

The six-term exact sequence that starts a minimal resolution of A(0) over A(1) determines a class in Ext4 (1,2) (A(0), 1F2). By constnicting a minimal resolution of F2 over A(1), the reader will find that H3,11 (A(1)) {0 } 4,11 (A(1)), and that the isomorphism at bidegree (4,12) is induced by Hthe

quotient 77: A(0) —> F2. If 0 <— F2 <- Q. is a minimal resolution of F2 over A(1), then there is a morphism of exact sequences: 0 <— A(0) <— A(1) <— P, <— P2

P3

s l2 A(0 )

Q3

Q4

ri

0

F2

A(1)Q,

Q2

s,2,2

0

436

9.

The Adams Spectral Sequence

s12-2 It' denote the map representing the unique class {u} G To complete the proof of the lemma, we show that right multiplication by the cohomology class of u determines the isomorphism given by left multiplication by the class of the six term short exact sequence given by the minimal resolution of A(0) over A(1). A minimal resolution of A(0) can be constructed from the six-term exact sequence,

Let U:

Q.

B-4,12(ACI)) NN .

0 <— A(0)

p2

A(1) <— P,

p3

s l2A(1)

s l2pi

Given a class in Ext A( s 't 1) (A(0),1F 2 ), it may be represented by a cohomology class of a homomorphism f: Ps —> stF2. Lift this mapping through the minimal resolution 0 <— Q., suspended t times. The commutativity of the diagram p +4

s' 2

5

P8

1.912f

f41

s t+12 F2,

8 t(24

s u

where vs is the isomorphism that carries the periodicity of the resolution of A(0) over A(1), follows by the properties of the resolutions and liftings. The different composites imply that multiplication on the right by {u} E EXt (112) (2 ) is isomorphic to left multiplication by the element determined by the periodicity of the resolution. ,

We next generalize the isomorphism of Lemma 9.49 to the other subalgebras A(n) of A2. Our point of departure is the observation that H4 ' 12 (A2) = { 0} _ R-5,1202 N) , while H4,12 (A(1)) F2. More generally, we observe that, for r > 3, we have hr+ ,h3 - = 0 in H"(112): Consider the composite of Steenrod operations acting on H" A( 2 ) : sq2'sq2' 1.. ,S'q2,S'q l (hi ho ). By Theorems 9.10 and 9.36, one finds that, for r> 2, Sq 2 ' • • • Sq2 Sq l (hiho) = hr+,hr•

Since h3h3 = 0, it follows that hr+ihr =0 for r > 3. These relations suggest the following construction: The dual of the inclusion j: A(r) A2, annihilates "12:'+1 , the class dual to Sq 2 '+1 . Let Cr be a class in Fs (A"1 ) such that d5 (Cr ) = [6. • • •

er l i,

2' times

which represents hr± i. The class e Cr in Fs (A(r) dual,) is a cycle. We let cor = {C r } denote the cohomology class of i*c.„ in H2' ,3' 2'(A(r)). 1Adams661 described explicit representatives for the classes cur using the — 1 product on the cobar construction. We need the following facts from this explicit construction:

9.6. Further structure

437

Fact I. co2 in H4 ' 12 (A(2)) maps nontrivially to the class {0 in H4 ' 12 (A(1)) that induces the isomorphism in Lemma 9.49. To see this, consider the short exact sequence of finite Hopf algebras,

0 —> A(1) —> A(2) —> A(2)//A(1) —> O: A(2)//A(1) is an exterior algebra that is 3-connected, and so the associated long exact sequence provides an isomorphism H4,12 (A(2)) B-4,120(1)). JAdams661 gives a nonzero class for W2 and so it must go over to the unique generator for H4,12 (A(1)). Fact 2. Under the inclusion A(r) —> A(r +1), the class wr+i maps to (w r ) 2 in cohornology, H*(A(r +1))

H*(A(r)).

This also follows from the explicit representative given by [Adams661. The cobar classes Cr may be used to define Massey products; let ker(h8') denote the subset of Ext ;8 tt2 (L,12) of elements whose product (on the right) with vanishes. In the cobar construction, Ldual let a represent a class a = {a} G Ext,8,it2 (L ,F 2) satisfying F:(11 0:21ulale y be such that d*y = a[6 • • • 6]. Define the homomorphism

hr

: ker(hr )

EXts-42' A2

,t+3.2'r (L, 72)

ExtsA+2"- 1,t+2' (L, 72) hr±i

by Pr—i (a) = {acr + ye 1 ]}. Notice that Pr—I is well-defined because the choices are absorbed into the quotient. Since Cr is a specific choice of element with d* (Cr) = [6_ 6_ • • • C2' ' ] this class PT-1 (a) can be further projected to Ext 2 ,t+3.2 '(L, 1 2) modulo indeterminacy where the indeterminacy is given by

s 2±2'+1 (L, 12)123 + Ext /

EXt A s +2

27'-1,t+2'

(L, 1 2)hr+1.

Thus Pr'(a) represents the Massey product (a, hr, 12,± 1). Consider the mapping

s ) (L,F 2 ) — 0 wr : Ext 4

Ext As+0„2;'t+3.2r (L,7 2 ),

given by the composition product on the right with wr . Lemma 9.51. The following diagram commutes: ker(hr)

EXt1+2' ' t+3.2' (LI

1 2)

Ext8+2'—l't+2'(L' .42 12)hr+1

Ext „sit(r (L,12)

s+2' ) Ext A(r)

(

L, 12)

9.

438 PROOF:

The Adams Spectral Sequence

i* (ac + y[62:+1 ]) = (i*a)(i* cr ) = (i* a) o Wr.

111

This shows that the mapping — O c4),„ is computable in terms of Massey products. This identification will be sharpened as we proceed. Theorem 9.52. If L is an (n —*connected module over A(r) that is free over r) (L,72) —> Ext As+(r2;''+3.2' (L,7 2) is an isomorA(0), then — o Ext phism for s > 0 and t
PROOF: For s = 0, the result follows because the Ext groups vanish in these degrees. We proceed by induction on s and t — n. Consider the short exact sequence 0 —> K —> A(r) A(1) L —> O. We make some useful observations: Fact 3 (A change-of-rings theorem). If H is a sub-Hopf algebra of a cocommutative Hopf algebra F over afield k, then r is free as an algebra over H, and furthermore, for all r-modules L,

Extr (r ® H L, k).

Ext t (L, k)

Fact 4. If L is an A(r)-module that is free over A(0), then, for r < p < oc, A(p) ® A (r) L is free over A(0).

Fact 3 follows from a theorem of [Milnor-Moore65]. Fact 4 is proved by [Adams66, p. 368 1 from an explicit choice of representatives for the dual comodules. Since L is free over A(0) and, by Fact 4, A(r)0A( i )L is free over A(0), we have that K is free over A(0). Notice also that K is (n+3)-connected. Consider the commutative diagram, where we have written Hs , ' (M) for Ext ,s t(r) (M, 11 2): B-s—i,t(A( r ) ®A(1) L

B -s—i,t ( K )

)

B-s,t ( L )

(0,1, Hs+2'-1,t+3.2'(A( r )

A(1) L)

Trs+2'-1,t+3.2'(K)

_El

Hs,t(A( r) 0") L)

Hs+2t+3.2'

(A(r) 0A( 1 )

,

L)

H5+2t+3•2'

Fist ( K )

Hs+2t+3.2'(K)

By induction, we assume the results hold up to s — 1 and for all (N — 1)connected modules and t — N < 4(s — 1). The first and fourth vertical arrows are isomorphisms by applying the change-of-rings to get Hs , ' (A(r)0 A(1)L)

9.6. Further structure

439

Ext s't (L T2) This isomorphism takes wr to (w2) 2" 2 , which is an isomorphism for t — (n+4) G 4(s — 1) by Lemma 9.49. Since K is (n+ 3) -connected, the second and last wr are isomorphisms for t — (n + 4) < 4(s — 1), that is, t — n < 4s. The Five-lemma implies that the third cor is an isomorphism and the theorem is proved. In order to extend this result about subalgebras of A2 to the entire Hopf algebra, we need the following approximation result.

Theorem 9.53. Suppose r