London Mathematical
Society Lecture
Note Series.
18
A Geometric Approach to Homology Theory by S.BUONCRISTIANO, C.P.ROURI(E, and B.J.SANDERSON·
CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON . NEW YORK . MELBOURNE
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Contents
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©
Cambridge University Press
1976 Page
Library of Congress Catalogue Card Number: 75-22980
Introduction
1
I
Homotopy functors
4
II
Mock bundles
19
Printed in Great Britain
III
Coefficients
41
at the University Printing House, Cambridge
IV
Geometric theories
81
(Euan Phillips, University Printer)
V
Equivariant theories and operations
98
VI
Sheaves
113
The geometry of CW complexes
131
ISBN: 0521 209404
VII
Introduction
The purpose of these notes is to give a geometrical treatment of generalised homology and cohomology theories.
The central idea is that
of a 'mock bundle', which is the geometric cocycle of a general cobordism theory, and the main new result is that any homology theory is a generalised bordism theory.
Thus every theory has both cycles and cocycles;
the cycles are manifolds, with a pattern of singularities
depending on the
theory, and the cocycles are mock bundles with the same 'manifolds' as fibres. The geometric treatment, pl
which we give in detail for the case of
bordism and cobordism, has many good features.
easy to set up and to see as a cohomology theory. transparent
Mock bundles are
Duality theorems are
(the Poincare duality map is the identity on representatives).
Thorn isomorphism and the cohomology transfer are obvious geometrically while cup product is just 'Whitney sum' on the bundle level and cap product is the induced bundle glued up. Transversality the geometric interpretations
is built into the theory -
of cup and cap products are extensions of
those familiar in classical homology. Coefficients have a beautiful geometrical interpretation and the universal coefficient sequence is absorbed into the more general 'killing' exact sequence.
Equivariant cohomology
is easy to set up and operations are defined in a general setting.
Finally
there is the new concept of a generalised cohomology with a sheaf of coefficients (which unfortunately does not have all the nicest properties). The material is organised as follows.
In Chapter I the transition
from functor on cell complexes to homotopy functor on polyhedra is axiomatised, the mock bundles of Chapter II being the principal example. In Chapter II, the simplest case of mock bundles, corresponding to pl cobordism, is treated, but the definitions and proofs all generalise to the more complicated setting of later chapters.
In Chapter III is the geo-
metric treatment of coefficients, where again only the simplest case,
1
pl bordism, is treated. is given in this case.
A geometric proof of functoriality for coefficients
NOTE ON INDEXING CONVENTIONS
Chapter IV extends the previous work to a generalThroughout this set of notes we will use the opposite of the usual
ised bordism theory and includes the 'killing' process and a discussion of functoriality for coefficients in general (similar results to Hilton's treatment being obtained). In Chapter V we extend to the equivariant case and discuss the Z 2 operations on pl cobordism in detail, linking with work of tom Dieck and Quillen.
Chapter VI discusses sheaves, which work
nicely in the cases when coefficients are functorial (for 'good' theories or for 2-torsion free abelian groups) and finally in Chapter VII we prove that a general theory is geometric.
The principal result is that a theory
has cycles unique up to the equivalence generated by 'resolution of singularities'.
The result is proved by extending transversality
to the
convention for indexing cohomology groups.
This fits with our geometric
description of cocycles as mock bundles - the dimension of the class then being the same as the fibre dimension of the bundle.
It also means that
coboundaries reduce dimension (like boundaries), that both cup and cap products add dimensions and that, for a generalised theory, n
h (pt.) ~ h (pt.). However the convention has the disadvantage that n ordinary cohomology appears only in negative dimensions. If the reader wishes to convert our convention to the usual one he has merely to change the sign of the index of all cohomology classes.
category of CW complexes, which can now be regarded as geometrical objects as well as homotopy objects.
Any CW spectrum can then be seen
as the Thorn spectrum of a suitable bordism theory.
The intrinsic geo-
metry of CW complexes, which has strong connections with stratified sets and the later work of Thorn, is touched on only lightly in these notes, and we intend to develop these ideas further in a paper. Each chapter is self-contained and carries its own references and it is not necessary to read them in the given order.
The main pattern of
dependence is illustrated below. I
t
7~T IV
V
VI
~ VII The germs of many of the ideas contained in the present notes come from ideas of Dennis Sullivan, who is himself a tireless
cam-
paigner for the geometric approach in homology theory, and we would like to dedicate this work to him.
2
3
It then follows that
I·Homotopy functors
(3)
if
a,
T E K, then
Notice that we do not assume A subset and we write
L
C
E K.
K is a subcomplex if L is itself a ball complex,
(K, L) for such a pair.
If (K , L ) is another pair, and o 0 then there is the inclusion
K
The main purpose of this chapter is to axiomatise the passage from functors defined on pl on polyhedra.
cell complexes to homotopy functors defined
Principal examples are simplicial homology and mock
bundles (see Chapter II). Our main result,
is a union of balls of K.
anT anT
C K, L cLare subcomplexes, o 0 ---(K , L ) C (K, L). An isomorphism f: (K, L)'" (K , L ) is a pl o 0 1 1 homeomorphism f: IKI •.• fK I suchthat fiLl = IL I, and a EK 1
implies
f(a)
E K.
1
1
In the case where K and K
plexes, there are simplicial maps f: (K, L)'" (K , L). 1
3.2, states that the homotopy category is iso-
morphic to the category of fractions of pl formally inverting expansions.
cell complexes defined by
K x L of ball complexes The categories
K, L is defined by K x L =
----
{axT
I uEK,
TEL}.
Bi and Bs
Now define the category
Bi to have for objects pairs
Analogous results for categories of simplicial complexes have been
morphisms generated by isomorphisms
proved by Siebenmann, [3].
morphism is an isomorphism onto a subpair).
In §4, similar results are proved for A-sets.
This gives an
alternative approach to the homotopy theory of A-sets (compare [2]). In §6 and §7, we axiomatise the construction of homotopy functors Here we are motivated by the coming applica-
tion to mock bundles in Chapter II, where the point of studying cell complexes, rather than simplicial complexes, becomes plain as the Thom isomorphism and duality theorems fall out.
(K, L) and
and inclusions (i. e., a general The category
Bs has the
same objects but the generating set for the morphisms is enlarged to include simplicial maps between pairs of simplicial complexes. Subdivisions If L', L are ball complexes with each ball of L' contained in some ball of L and L'
IL' I =
The categories
IL I, we say L' subdivides
L, and write
Bi and Bs enjoy a technical advantage over
categories of simplicial complexes;
The idea of using categories of fractions comes (to us) from [1] where results,
The product
1
Thus to define a homotopy functor, it is
only necessary to check that its value on an expansion is an isomorphism.
and cohomology theories.
1
are simplicial com-
then there is a complex L'uK=
namely, if L
{a:uEl'
C
K and L' <J L,
or K-Ll.
similar to those contained in §4 here, are proved.
Throughout the notes we use basic pl
concepts; for definitions
and elementary results see [4, 6 or 8].
Collapsing We assume familiarity with the notion of collapsing,
as in [6],
1. DEFINITIONS
for example.
Suppose (K , L ) C (K, L), where L = L n K ; then we o 0 0 0 have a collapse (K, L)' (K , L ) if K '" K and any elementary col-
Ball complexes
lapse in the sequence from a ball in L is across a ball in L (so that
----
Let K be a finite collection of pl IKI = u {a:
4
a
E K l.
n balls in some R , and write
Then K is a ball complex if
(1)
IK I is the disjoint union of the interiors
(2)
a E K implies the boundary
a
a of the
in particular, expansion.
0
0
0
L'
L). We call the inclusion (K , L ) C (K, L) an o 0 0 The composition of an expansion with an isomorphism is still
called an expansion. a E K, and
is a union of balls of K.
5
2. SUBDIVISIONIN THE CATEGORYOF FRACTIONS
J
Let B = Bi or Bs, and let 2.; denote the set of expansions. B[L-I]
category of fractions
;/~
The
is formed by formally inverting expansions.
Thus the objects are the same.
New morphisms
e -1, eEL,
K f~
duced, and a morphism in the category of fractions is then an equivalence class of formal compositions g. = e~1 for some e. E L. 1
1
g
0
1
go 2
...
g , where g. E B or
0
n
1
The equivalence relation is generated by the
1
commutes.
following operations: (i)
replace
(H)
introduce
(iii)
replace
h by fog e
0
(e e) 2
1
e
-1
-1
if h = fg and f, g E B; or e by e
-1 0
-1-1 0
1
Proof.
e, eEL; e
2
.
In fact operation (Hi) is a consequence of operations (i) and (H). Denote the equivalence class of a formal composition by {g The category of fractions is characterised ping property;
L
A
are intro-
go ... g }. a 1 n by a universal map0
K~J.
0
namely, given any functor F: B -+ C such that F(e)
is an isomorphism for each eEL,
Define J = K U L, where J is regarded as a suba J complex of both by e and f. Then let e and f be the obvious ina a elusions. That e is an expansion follows by echoing the collapse a
then there exists a unique functor
F'
Remark.
The lemma fails for Bs.
K = I x I J = I O} x I and L = I 0 }. Then f must be degenerate on " a eJ and hence be degenerate on the 2-cell in K.
so that
Now suppose L'
B
a morphism denoted
[>
Ie -1
L' x {I}.
U 0
i) : L
commutes, where p is the natural map. For simplicity,
in the rest of the paper we will ignore pairs
(K, L) with L"* p' when the general case can be obtained by making
We first observe that any morphism in Bi[L-I]
minor adjustments. in fact be written
{e -1
Lemma 2. 1.
0
f)
may
by repeated use of the following lemma.
Let e : J -+ K be an expansion and f : J -+ L ~
mo!p~!sm i~ Bi. Th.~_~!1e_:r:~~ _~?:p~xpa.~~!on eo and morphism that
fa so
Then e is an expansion, and so we have
L' in B[L-I],
-+
i: L-+LXIuL'x 11 }
called subdivi~lon and
(L, L').
Lemma 2. 2.
C>
(L, L)
is function~~_!ha~~~!
=
(i)
C>
(H)
if L"
[) (L', L")
0
[>
idE!~ti!y~
(L, L')
= C>
(L, L").
Proof.
For part (i), we must show that if i , i : K -+ K x I a 1 are the inclusions, then {i } = {i ). This is proved by simple cola 1 lapsing arguments. First, consider K x I subdivided to (K x I)' by placing K at K x
H 1.
(K x I)' and a reflection Now ri.!.
=
i.!., and
222
It follows that
II'}
There are then inclusions 1':
{i.!.} is an isomorphism since
=
i~, i~, i~ of K in
(K x 1)' -+ (K x I)' about the half-way level.
identity.
Since ri'
The result then follows by considering
6
For instance, take
= il,
a K x t::.2•
(K x I)' '" K x {~ }.
Ii'} = {i' 1. a 1 This argument is
we have
7
(a x I) u a" x {1 } '"
essentially due to Siebenmann [3; p. 480]. For part (ii), let .
Let (L x
Il , I:. , 1:..
o Then (L x
1.
2
be a 2-simplex with vertices
1:.
111
and OpposIte faces L" x {v
2
1
2
1:.2)'
'X
L'
X III
U 0
L
The proof is completed by a diagram chase.
2
Il )'
=
L x
1 X 1:. U 1
I:.
2
x I) U (J x
{o } U a"
c
x {1 } '" (aX I) u a x {0 } ,
v , v , v 012
UL'
x
Il
1 0
where the first collapse is elementary,
U
L" x {v }~ L" x {v ) 2
Corollary 2.4.
2
See Fig. 1.
and the second is cylindrical.
Suppose L'
(L, L') :L-L'
is
an isomorphism. Proof. so that
L"
[>
It follows fr om 2. 3 that there is L"
(L, L") and
[>
(L', L'") are isomorphisms.
The result now
follows from 2. 2.
•
<'I;
(a
~
3. ISOMORPInSM WITH THE HOMOTOPYCATEGORY Now let Bh denote the category with objects pairs of ball complexes, and morphisms homotopy classes of continuous maps.
Then there
are natural maps Bi
BS---n -----
t
~ L'·-
Bh
and by the universal mapping property, we have a diagram Fig. 1 Lemma 2. 3. such that (L x I) Proof.
U
Suppose given L'
L"
l.
Let aI' ... , an be the balls of L listed in order of
decreasing dimensions.
We subdivide L in n steps.
the interiors of a , ... , a
r
1
are not touched again.
After step r,
Suppose r-steps
Theorem 3.2.
Proof.
a
such a a
c a',
T.
Then a c
Subdivide
c simplicial.
a"
=
T
=
subdivision of a. Now assume there is
a - interior
(T)
is a collar on
a
in a, and
to a" so that a collar projection a" - a" is c c c U a~ is the required subdivision of (J. Note the (J
a" ~ a". c The resulting complex L" clearly has the desired property since
cylindrical collapse
8
of
categories.
completed, and let a= a +l• Then a has been subdivided to a', say. r Let T £ a' be a cell with dim T = dim a, and T n = ¢. If no such T exists, perform a preliminary
The maps in diagram (3,1) are isomorphisms
Since all three categories have the same objects, it
suffices to show that each of
0,
f3,
y is an isomorphism
morphisms from K to L for any K and L.
on the set of
We prove this in three
steps: A.
0
is surjective,
B.
0
is injective,
C.
f3 is surjective.
9
The result then follows by commutativity. y
are compatible with subdivision; Remark 3.3.
f
We first observe that a and
1.e. ,
then by the pl
c
(K x I)'
K
Suppose [f]: K ...•L is a homotopy class; mation theorem,
0
Suppose L' <J L; then a(e> (L, L')) is the homo-
a is surjective.
A
f
'
.••
1
1
Commutativity in Bi[~-I]
L
J'--<J g
:=J
J
0
II
7 !~
M
,.
f
approxi-
* there exist subdivisions K'
••
J
K~
topy class of the identity map. ~tep ~
e
,.
0
/I
e
L 1
follows from definitions and 2.2.
simplicial Ii1ap f' : K' ...•L' so that [f'] = [fl. Let M(f') be the sim-
(Note that the left-hand half commutes by 2.2(0.)
plicial mapping cylinder of f'.
morphism by 2. 2 and 2. 4.
There is then an inclusion i: K' ...•M(f'),
and an expansion e : L' ...•M(f'). (L, L'fl{e-1Qil[)
[f]=a([)
Step C.
(K, K')).
{3 is surjective.
It is sufficient to show that the class in Bi and simplicial maps.
J
yO~
i
L J1
U i 1
are expansions, and such that a {e-1
0
f ) = {e
0
11
fl.
-1
Now a {e
0
00
f
0
0
1=
1=
f
a {e~1
0_1
a {e
0
11
f
means that the diagram homotopy commutes, provided we regard and e
1
0
f1
l.
~
r;
M
f'
f
Ue
K
•
L
f
where Mr is the simplicial mapping cylinder of f, and (K x 1)' is
eo
K x I derived at K x {~l.
Now let J = J u J. Then by homotopy commutativity and the o L 1 relative pl approximation theorem (see footnote ), there is a sim-
(K x I)' ...•J' so that f I /KI x Ji 1 = f.,1 i =
inclusion
1
1
as homotopy equivalences.
plicial map f :
o
(K x 1)' -
~~1
00
Consider the commutative diagram
K
K
we have {e
{f 1 of a simplicial map
f : K ...•L is in the image, since any map in Bs is a composition of maps
By 3.1, it is sufficient to show that for any diagram
_11
The map g is an iso-
The result follows.
It follows from 3.3 that
step B. a is injective.
in which the e.
L
0, 1.
Now
obvious vertex map. {f)={e
-1
of'oi
1
The simplicial map f' is defined using the
Then i l={e
_10
and e are expansions, otoi
-1 0
oi
1
and we have
-1-1
otoi
)={3{e
0
oi
1
1,
since each map in the bracket lies in B1.
consider the following diagram in which arrows marked e are expansions, and arrows marked
[)-+
4.
are compositions of subdivision followed by
inclusion:
A-SETS AND THE CATEGORY OF FRACTIONS We assume familiarity with the basic definitions involving A-sets
which are found in [2]. An inclusion of A-sets
*
This is weaker than the usual simplicial approximation theorem. [4] or [8] for a short proof. 10
See
L CK
is called an ele-
ment~y exp~nsion if K is obtained from L by attaching a set of sim-
11
plexes
~n(s), for s in some indexing set, to L via Li.-maps
f s : A11,1,is) - L. An inclusion countable sequence that K =
U
L,.
1
L c. L
L
C
In particular,
C
{X, K} has a representative an expansion.
_
of elementary expansions so
the 'end inclusions' of K in K
easily seen to be expansions.
We first show that l/I is surjective.
L ...
n'
L
C 123
Proof.
K is an expansion if there is a -... @
I are
Let L: denote the set* of Li.-maps which
roe {e-I
= id.
e -I
0
f, where
An element of
f is a Li.-map and e is
Using the Kan condition on K, we find a Li.-map r so that
Define g = r
0
f.
{g) = {r
Then
0
f) = {r
e 0 e- 1 0 f ) =
0
fl.
0
are expansions, and let ~[L.;-I] denote the resulting category of fractions. The analogue of 2.1 is easily proved for Li.-sets so that any morphism in ~[2:;-I] may be written
{e-1ofl.
Let ~h denote the category of Li.-sets
and homotopy classes of continuous maps between realisations. Theorem 4. 1.
X
----
.• ,-
Thus l/I[g] = {e-I that lJI[go]= lJI[g)
Proof.
The proof is analogous to the proof of 3.2.
The main
K
g
1 The canonical functor Li.(L:- -) Li.h is an iso-
.f l.
0
To see that
This means that gl is obtained from
(i)
replace
(fg) by fog,
This can be done by deriving.
(ii)
introduce
(iii)
replace
e or e e-I , (e e fl by e-I e-I
Also, the Li.-sets need to be replaced by
simplicial complexes before applying the simplicial approximation
t
This is done by using (L x 1)' with L x {1)
and L x {~} derived once.
derived twice,
We leave the reader to check details.
Now let K be a Kan Li.-set, and let [X, K] denote Li.-homotopy classes of Li.-maps XX- K in ~(L:-I). l/I: [X, Kl- {X, K).
K; let
{X, K} denote the set of morphisms
From Theorem 4. 1, we have a well-defined function In fact, l/I can easily be well-defined without the aid
of 4. 1, and the proof of the following theorem is then independent of 4. 1. Theorem 4.2.
Suppose X is a Li.-set and K is a Kan Li.-set.
Then the canonical function l/I :
go by a
e-I
0
0
0
2 1
1
2'
Now it is easy to see that the map g defined above is unique up to Li.-homotopy; use the expansion L x {O) U K
@
I U L x {1) -L
@1.
Hence after each step (ii), we can map each A-set in the composition into K uniquely up to Li.-homotopy. It follows that g up to Li.-homotopy.
o
and g
1
are the same
Now let CWh denote the category of CW complexes and homotopy classes of maps. -I
f : ~[L:
Then there is the composition
] - ~h - CWh.
Theorem 4. 3.
f : ~[L:
-I
] - CWh is an equivalence of categories.
[X, K] ....• {X, K}
Proof.
It is sufficient to show that if X is a CW complex, then
I S(X) I ....•X
the natural map
is a bijection.
we have an isomorphism * P. May has pointed out that there are set theoretic problems here. We adopt MacLane's axiom of one universe [7; p. 22]. Theorem 4. 1 then shows that ~[L:-I]
is a category in the universe.
t The required approximation theorem is given in [2; 5. 1].
12
suppose
sequence of the following steps and their inverses:
modifications are to replace the cell subdivisions used hy simplicial ones.
theorem,
is injective,
lJI
a
1T
nSX -
fJ
is a homotopy equivalence. 0
From definitions,
n:
f3
1T
n Isxl -
1T
nX.
But a is an isomorphism by a relative version of 4. 2. It follows that f3 is an isomorphism,
and the result follows from J. H. C. Whitehead's
13
defined to be equivalent to u
theorem [5J.
1
E
T(K)
Remark 4.4.
There are easily provable relative versions of the
-
=
T(P)
1
if there exists
1
some K' <J K., such that am(u.) 1
=
=
u, i
u
T(K') for
E
0, 1. Alternatively,
lim T(K) where the limit is taken over all K with
The canonical function T(K) - T(P) is of course a bijection.
theorems presented in this section.
=
IKI
P.
If
[f) : P - Q is a homotopy class, then T[f] is defined by choosing a 5. HOMOTOPYFUNCTORS
representative
Finally, note that for an arbitrary
We rephrase Theorem 3. 2 in terms of functors defined on B = Bi or Bs.
category
T : Ph - C by choosing for each P a particular
Let T : B - C be a functor. L et e : (K0' L)0 ....•(K' L) be an expansion.
Ax C (C011apse.) _lOm __
for [f) in Bs.
Then
T(e) is an isomorphism. From the universal property of the category of fractions and
then defining T(P)
=
C, we could define
I K I = P,
K with
and
T(K). Any two such choices give naturally equiva-
lent functors. 6. CONSTRUCTIONOF FUNCTORS
Theorem 3.2, we have Suppose given a contravariant functor Theorem 5. 1.
Any functor T satisfying C factors uniquely
through Bh; moreover,
if T is defined on Bi, then it extends uniquely
to Bs. Suppose now T is contravariant the canonical extension by T also.
and satisfies Axiom C. Denote
Suppose
IL I =
IK I; then we have
following axioms. E
(extension). Suppose e : (K , L ) - (K, L) is an expansion; o 0 Z(e) : Z(K, L) ....•Z(K , L ) is surjective. o 0
G (glue).
Suppose K = K UK, 1
Suppose u.
an isomorphism
E
1
=
L
Z(K., L.), i 1
1
L n (K n K).
o to u., i
T[id] : T(L) - T(K).
Z : Bi - S* satisfying the
1
1
=
=
2
L
C
K, and L. = L n K., i = 1, 2. 1
1, 2, restricts
Then there exists
2
1, 2. Moreover if K n K 1
2
then
z C
to u E
E
1
Z(K
1
fI
K , L ), where 2
0
Z(K, L) so that z restricts
L then z is unique.
Define T(K, L) = Z(K, L)/-,
If L <J K, we call this an amalgamation isomorphism and write z
E
zI
am : T(L) - T(K).
where z _ z if there is a o 1 Z(K x I, L x I) so that z. is identified with the restriction 1
E
Z(K xli},
L x Ii} ~ Z(K, L). It is an easy exercise to show
that - is an equivalence relation. From definitions, we see that am = T(I> (K, L)). The inverse isomor-
Z(K x ~ , L x ~ ). 2
phism is called a subdivision isomorphism and we write
To see that z - z, consider
2
Now if f : (K , L ) ....• (K , L ) is a morphism in Bi, then
sd : T(K) - T(L).
1
1
2
2
T(f) : T(K , L ) - T(K , L ) is clearly well-defined by 2
2
1
1
Now let Ph be the category of compact polyhedra and homotopy classes of continuous maps.
Let T : Bi ..• S* be a functor satisfying
Axiom C, where S* is the category of based sets.
We define T: Ph -S*
as follows. An element of T(P) is an equivalence class of elements of T(K), where
IKI = P.
Suppose IKil = P, i = 0, 1. Then
Uo
E
T(Ko)
T(f)[z] = [Z(f)z]. Proposition 6. 1. Proof.
The functor
T satisfies Axiom C.
Let e : (K , L ) - (K, L) be an expansion. o 0
We have to
is 15
14
show T(e) is an isomorphism. Suppose then that T(e)z
= T(e)z. Construct o I in two steps. First, find z
that z : z _ z o I using Axiom G twice; then find z using E.
z E:
2
E:
Z(K x I, L x I) such
Z(KU(K XI), L U(L xI» 0 a
K=>K,K: I
T satisfies the following axioms for any
2
Half exact:
T(K, Kl UK)"'" T(K, KI)"" T(K2,
Kl n K)
is exact. Excision:
Compatibility of extension to Bs Now suppose that Z is in fact defined on Bs.
Then we have T
defined using Z I Bi, and T extends uniquely to Bs by Theorem 5. 1. Proposition 6. 2.
Lemma 7. 1.
But by Axiom E we have T(e) is onto.
T(K UK, I 2 is an isomorphism. Proof.
Order 2 is obvious; to see exactness,
a concordance on K
The extension of T to Bs is given by
K ) ...•T(K , K n K ) I 2 I 2
2
use E to extend
to one on K. For excision, use the definition of
Z(K, L) and G. T(f)[z] Proof.
=
[Z(f)z].
Now suppose given functors
By uniqueness,
it suffices to show that T(f) is well-
defined on Bs by the above formula, case f simplicial.
and it is sufficient to consider the
zq for q
E
Z' defined on Bi and
extended to Bi as above, and suppose that in addition we have Axiom S (suspension). *
There are given natural isomorphisms
Let K be any cell complex, and define (K x I)'
by deriving each cell on the half-way level. so is (K x 1)'; and if f: KI - K
2
Then if K is simplicial,
is simplicial,
then the deriveds may
be chosen so that L(f x id) : (K x I)' ....•(K x I)' is simplicial. I
2
Then we can define
The
result therefore follows from Lemma 6.3. that z. = Z I K x {i l.
z o - z I if and only if there is z
E:
Z (K x I)' such
1
Proof.
Consider
Q = (K X ~2),
to be the composition sq Tq(L) ...• Tq-1(L x I, Lxi)
0
1
E:
Z(K
X ~2)"
it is easy to see that Q col-
{o} )
TO) L)
2
lapses to both these subsets. "i
Moreover,
Tq-1(W, KULX
Tq-1(K,
I
and (K x (~l U~l»,.
I
+-
obtained by deriving each cell
of K x {v }. Then Q contains isomorphic copies of (K x 1)'; namely, K x ~l
T(i)
and hence
"il
Therefore, E
given a z
E:
Z(K x I), we get
Z(K x I)' and vice versa.
where W = K U x {1 }L x I, and i is an excision, j extends the identiL fication L - L x {O}, and as a map K"'" W is homotopic to the inclusion by an extension of the obvious homotopy on L.
7. COHOMOLOGYTHEORIES Now let Bi with L
=
C
Then easy arguments show that the long sequence is exact, and
Bi be the subcategory consisting of pairs
¢, and suppose that
Z : Bi - S* is a functor.
extend Z to Bi by defining Z(K, L)
=
(K, L)
we have shown
Then we can
Ker {Z(K) - Z(L) l. Suppose Z
now satisfies axioms E and G. Let T denote the associated homotopy
* See the note on indexing cohomology groups at the end of the introduction.
functor.
16
17
Theorem
7.2.
A sequence
sat~fying
E, G, and
S defines
~~~ll1pact
polyhe~al
pair~
Remarks
7.3.
sq~, sq."
E
sq~ + sqTj suspension
Zq-I(K E
Zq-I(K
to return
2,
i),
and use
to
In fact,
homology theory
Given
x I)'.
Finally,
~,
I g I "" n I g
1/ E
Z,
E
II-Mock bundles
on the category
Zq(K), form
G to construct
excision,
by formal
with
We describe and inverse
gk
=
example.
and suspension
of
imply co-
using Puppe sequences.
&I-spectrum
for
zq(~k)
11. q We will ~XPlain
qII q §5 for a specific
S*, q
theory pi
half exactness,
g
-+
Zq(K).
A classifying
taking ail-set
theory
use amalgamation
argument, q Axiom S need hold only for T (,). 3,
Bi
is in fact an abelian group functor.
addition':
x I, K x
zq:
a cohomology
Tq()
1.
This is seen by 'track
of functors
q T ()
Thus
can be constructed
by
in detail in Chapter 'f" Inl't e comthe theory t 0 In
REFERENCES
FOR CHAPTER
proofs.
proof of the
Quinn's
Another
~-sets
[2]
Homotopy theory and calculus
fractions.
Springer-Verlag,
Berlin
C. P. Rourke
and B. J. Sanderson.
Oxford Ser-, 2 (1971),
p]
L. Siebenmann. Akad.
J. Stallings.
[51
J. H. C. Whitehead.
[61
are constructed.
notes on polyhedral
Combinatorial
Seminar
Categories
Verlag,
C. P. Rourke linear
homotopy types.
Proc.
Kon. Ned.
topology,
homotopy 1.
Bull.
1969. Amer.
+
topology.
a
it the block over
of a pi
topology.
I. H. E. S.
-1
p ~ (u).
tive,
and then
Springer-Verlag,
~(a)
=
¢ if dim a
empty set for total space, Figure
Berlin
is similar
to
a
<
E
-q.
K.
with base
p ~ : E ~ -+
manifold
IK I
such
of dimension
-1
p ~ (a) by
~(a), and call
of any dimension;
Therefore,
thus
q could be nega-
The empty bundle bundle for all
~, 17/K are isomorphic,
1/(a) for each
written
h: E ~ -+ E 1/ which respects a
E
Now define the based of q-mock
We denote
~q/K
¢/K q
E
has the
Z.
bundle over the union of two I-simplexes.
homeomorphism
=
pi
and is a q-mock
2 shows a I-mock
is a pi
classes Introduction
In a final section,
bundle*
as a manifold
may be empty for some cells
for the working mathematician.
and B. J. Sanderson.
a short
a.
Ha)
h(~(a))
New York (1971).
1. 2].
projection
1
dim a, with boundary
bundles
~ ~ 1/, if there blocks;
i. e. ,
K. set
over
Zq(K)
to be the set of isomorphism
K with base point the empty bundle.
to piecewise-
(1972). * The terminology
18
E ~ consists
A q-mock
K, P~ (a) is a compact
E
Mock bundles
on combinatorial
notes (1963-69).
S. MacLane.
[8J
J. Math.
[12;
can be com-
also provides
The construction
K be a ball complex.
for each
55 (1949), 213-45.
and Warwick
Springer-
1. Quart.
theorem
The empty set is regarded
simple
Tata Institute
E. C. Zeeman.
[71
~-sets
q
73 (1970), 479-95.
[4]
Math. Soc.
(1967).
that,
321-8.
Infinite
(Amsterda!!!1
of
The theory
have short
[7; §1].
Let
and M. Zisman.
theorems
is that mock bundles
case is have simple
MOCK BUNDLES AS A COHOMOLOGY THEORY
I
P. Gabriel
feature
transversality
which in the simplest
and duality
a COhomology transfer. pi
This is a bundle
all the usual products
transparent
K and total space [I]
theory
and the Thorn isomorphism
1.
plexes.
to a cohomology
definitions,
classifying
this construction
of mock bundles.
In this interpretation,
posed yielding
then it can be seen that
This extends
giving rise
cobordism.
here the theory
'mock bundle'
is due to M. M. Cohen.
19
by E(~) = E(~ ) u E(~ ), and define the projection of ~ inductively, h 1 2 using the fact that cells are contractible. For Axiom S (suspension),
s~1K x I by E(s~) = E(~) and Ps~ = p ~ x K x I at the half-way level.
UK. Now form
suppose given
{i J;
i. e., place
Then s ~ is empty over K x
~ over
i (and over
L x I if ~ empty over L), so we have a suspension map
The inverse map is defined by composition with the projection 11 :
K x I -+ K; i.
-1
Fig. 2
S
1]
IK by E (s
e.,
-1 1])
given
1]
IK x I empty over K x i, define
= E( 1]) and p(s
-1 1])
=
7T 0
P 1]'
It is trivial to check
that s -1 is an inverse for s. Zq ()
functor on t h e ca t egory Blo of ball com-
becomes a contravariant
plexes and inclusions via the restriction:
~IL
then
Suppose given ~IK and L
is defined by E(~IL)=p~l(L),
(We use both notations
Finally, we have to check Axiom E (extension). K;
and P(~IL)=p~I:E(~IL)-+L.
E(~) and E ~ for total spaces,
venient. ) Now define a functor
C
etc.,
as con-
zq(,)
on the category
Bi of pairs of ball
In other words, Zq(K, L) is isomorphism classes
I: is cobordant to ~
"0
l'
1]1K x I, empty over
written
Tq(,),
0
By induction on the length of the collapse,
the case when the collapse is elementary Let J be the subcomplex
T.
canidentify
~ _ ~ 0
if there is a mock bundle
l' 1]
IJ
IJI X {OJ).
I
0-
from a free
is a ball, and we
Wethendefine
EW = E(~ o ) UE(~ 0 IJ) x I identified over E(~ 0 /J) = E(~ 0 IJ) x and let PI: = p(~ ) on E(~ ) and p(~ ) x id on E(~ IJ) x I. 0
0
Lemma 1. 2. q-mock bundle.
Suppose
0
IKI is a pl
n-manifold,
{O
J,
That
and ~IK
Then E ~ is an (n+q)-manifold with boundary
and in any
Proof.
(Compare [4;p.
142].)
say. We can then define a 'transverse restricting axioms E, G, and S of Part I,
0q J is a cohomology theory* on the cate-
gory of pairs of compact polyhedra.
Then
T.
~
is a 1
I I).
p~ (a K
IK x li J ~ ~i for i = 0, 1.
case we now prove: satisfies
across a cell
a-
(0-, IJI) with (IJI XI,
it suffices to prove
as in I. 6, by taking Tq(K, L)
of mock bundles empty over L, where
L x I, such that
zq(,) and hence, by I. 7. 2, {Tq(,),
0
In other words,
~IK so that
is given, we have to construct
is a mock bundle follows from Lemma 1. 2 below.
It is easy to see that cobordism is an equivalence relation,
Theorem 1. 1.
and ~ IK
,,0
bundles which are empty over L. We can now define a functor to be the set of cobordism classes
of
o
~o ~ ~ I K.0 face
complexes (as in I. 7) by defining Zq(K, L) to be the kernel of Zq(K) .•• Zq(L).
if K X K
collars of a~(T) star
the two transverse
J.
star' to x in E(~) by inductively
in ~(T) for
of x in E(~) is homeomorphic to {transverse
Let x EE(~); then xEint~(o-),
0-
<
T.
Then a neighbouthood
{neighbourhood in ~(o-)J x
The same construction holds for p(x) in K, and stars are abstractly
isomorphic,
hence homeomorphic.
But IK I being a manifold implies that the transverse star is a disc (P X Q lS a man if 0 ld at (x, y) if an d on1y if pOtlS one a x an d Q one
Proof of 1••1 For Axiom G (glue), suppose given ~l1K1' ~21K2' hO h·I:IKnK~I:IKnK Form~IK=KUK an an lsomorp lsm . "1 1 2 "2 1 2' 1 2 at y; see [6]). The result follows on observing that x E int (transverse * See the note on indexing cohomology groups at the end of the introductiOi saran t ) if d on1y 1of X Em. ° tiKI 0
d·
20
21
Amalgamation
The Thorn isomorphism We
above generalises
at once to give a Thorn isomorphism theorem for pl
be given where K'
Let ~qIK'
finish the section by observing that the proof of Axiom S given
a mock bundle over K by Lemma 1. 2, called the amalgamation of ~
block bundles (see [11] for definitions). Let ur ,1( be a block bundle; then we can give E(u) a ball com-
and written
plex structure in which the blocks of u are balls, for we merely have to
extending ~IK' x {D)
choose a suitable ball complex structure
with E(~) x 1. This is checked by induction over the skeleta.
on E(u). Then we can define
am( ~). To see compatibility with amalgamation as in
Chapter I §5, notice that the bundle 1]IK x I U K' x {O}
obtained by
via the proof of 1. 1 has total space homeomorphic Uniqueness
of collars is used to match product structures. Subdivision to be composition with the zero section and p((~)) = i
0
i: K'" E(u); i. e., E((~))= EW
p~.
subdivision
Proposition 1. 3.
is an isomorphism, an~J:nduces an isomorq q-r . phi~m~_called the ThOIl1_~omorphism, T (K)'" T (E(u), E(u)). Proof.
An inverse for <J>is given by composition with a pro-
UK and K'
Now let and 1. 1).
~'IK'
Examine the proof of existence of a
such that am(~') - ~ (which follows from Chapter I
The proof says consider
K"
division of K, is cylindrical in each ball less a smaller ball (see 1. 2. 3). Then K" x {D)
u K x I 'K
to TJ IK" x
u K x I and let
{D)
is defined skeletally,
jection for u.
x {I}
so that we can extend ~IK x {I}
~'= am(1] I K" x
and, on a typical cell, a
extending over the cylindrical collapse Remark.
In the next section, we show that the Thorn isomor-
initial collapse
phism is given by cup product with a Thorn class.
1
u
axIuax
E
But the extension
K is obtained by first
a , 1
a
and then over the
{I}.
It follows that we can inductively identify E(1]) with E(~) x I since E(1] I a" - . a 1) ~ E(TJ I a) x I can be identified with a collar on ~(a),
2. THE GEOMETRY OF MOCK BUNDLES We now give geometric interpretations theory Tq(,) defined in §1.
a x I '" a" - a
an -
{D)).
and 1](a x 1) = Ha) XI (by the proof of 1.1). to parts of the cohomology ment is again used to match product structures.
Addition
Theorem 2. 1. exists
~'IK'
A simple collaring arguWe have proved:
Suppose given ~IK and K'
such that am( ~,) ~ ~.
There is an addition in Zq(K, L) given by disjoint union; i. e. , E U + 7]) = E (~) U E (1]) and p + = pup . It is then easy to check that Remark. The reader can check that this proof of existence of q this coincides with the 'track ;dd7tion' ~defin:d in T (,) (1. 7. 3). To subdivisions is essentially that given in [9; p. 128]. see the group structure
directly,
observe that ~ + ~ - ¢ by letting
1J1K x I have E(1]) = E(~ x I) and
Suppose f: K .••L is a simplicial map between simplicial complexes, and ~;l,a mock bundle.
P1](x, t)=(P~(x), t), (p ~(x), 1 - t),
Induced mock bundles
t?:
t.
E(f#(~)) - - - ~ E(~) ~
K ------.
22
Then we can form the pull-back diagram
f
~p~ L 23
and it is not hard to show that f # (s) is a mock bundle, and that
Now if we subdivide corresponding to
1) so that the blocks of ~ are subcomplexes,
a
E
K, we have by 1. 2 the manifold
E(1) I
s(a».
then In
other words, we have a (q+r)-mock bundle is well-defined and functorial (compare [12; 2.3]). This means that q zq( ,) and, hence by Chapter I, T ( ,) is in fact defined on the larger category
Bs, and it follows from 1. 6. 2 that the notion of pull-back is
which we call
compatible with the pull-back as homotopy functor.
sq IK
and 1)r;1.. be mock bundles.
defined by E(~ x 1)
0
1). P
E(1)
The external product Let
s
= EW
Then s x 1)IK x L is
1).
p~!p~
EW
x E(1) and p(~ x 1) ~ p ~ x P1)' The blocks
K
of ~ x 1) are then products of blocks of ~ and 1). We thus get an This extends to give a transfer
external product
Remark. [s x 1/]
=
In the case that 1) is the identity id: L
11*[~]where
11: K x L
suppose K, L, and 11: (K XL)'
-+ -+
K is the projection. K are all simplicial.
-+
L, we have
To see this, Then 11#(~)
where p = Ps' We observe that the transfer is functorial i. e. where p and q are mock bundle projections
is a subdivision of ~ x 1). The internal product Suppose given ~qIK .;).*[~ X 1)], where
.;).: K -+
The transfer can be seen to be the composition of the following: q Define [~] U [1)] E T +r (K) to bi K x K is the diagonal map; i. e., subdivide Tr (E(m Thorn iso,;. Tr-s(E(V), E(V»2:.Tr-s(KXlq+s, K>4q+1)~~sP'Tr+q(K) and 1)r/K.
~ x 1) so that .;).(K) is a subcomplex, and then restrict
to .;).(K).
The internal or cup product makes T*(K) into a commutative ring with unit.
To see that the class of id: K ..• K is the unit, use the
remark on external products and the fact that and commutativity are easily checked.
11
0
.;).
=
id. Associativity
There are natural relative ver-
sions of both products which we leave the reader to formulate.
where E(s) is embedded in K x int(lq+s)
with normal bundle vs.
Alternative description of the cup product Proposition 2.2.
m U [1)] = P, p*[ s], where •
--
Let' sq, 1)rIK p
be mock bundles.
Then
= p 1) .
In other words: pull s back over E(1) and then compose.
The composition and the transfer We now generalise the composition used in §1 to give suspension and Thorn isomorphisms. Let
~qIK be a mock bundle (a block bundle with closed manifold
as fibre is a special case), and let 1)r IE (~) be another mock bundle.
24
Proof. that p : E (1)'
We may suppose that K is simplicial. -+
K' is simplicial,
and subdivide
Subdivide p so
~ to s' IK'.
Let
(K x K)' be the subdivision of K x K given by [14; Lemma 1] so that ~(K) is now a subcomplex.
Consider the following diagram.
25
Thorn and the Euler classes are natural and multiplicative.
We leave
the reader to check these facts. Properties
of the cup product and the transfer
We now give some properties which will be used in Chapter V. Proposition 2. 4.
The squares are pull-backs,
and K" is constructed as follows.
f'
-----.,-
Choose
Suppose given a pull. back i!iagr~!!!.. E(~)
1
a triangulation of K x E(1]) so that the blocks of id x P'1/ are subThen choose a further subdivision, and choose .K" <1 K' so
complexes.
that 1r is now simplicial. 1
____
f
P
._
L
Then choose ~";K" subdividing ~';K'.
Without loss of generality, we can assume p~, = p~". Now P~ x P'1/ is
with f simpH~~al.__Then f*p!
=
PI f'*.
the projection of a mock bundle U(K x K)', since by construction
I; = 1r#(1]) 1r#(~"). Butfrom definitions P, p*[~]= A*[1;] = A*[~x1]]=[ ~]u[~
Proof.
Subdivide so that p : E( ~)' ...•K' is simplicial,
then sub-
0
2
1
.
The Thom class and the Euler class of a block bundle Let ur;K be a block bundle with i: K ...•E(u) the zero section. If E(u) is given the ball complex structure of §l (in which blocks are balls), then i is the projection of a mock bundle, and thus determines a class t(u) E T-r (E(u), E(u)), called the Thom class of u.
divide so that f : L' ...•K' is simplicial.
Now E(f#~) is a linear cell
complex in a natural way and we may subdivide without introducing new vertices so that p', f' are also simplicial.
The result now follows from
definitions. Remark 2. 5.
I KI
In the case
=
M is a manti old, f is the
projection of a mock bundle 1];K+, where K <1 K+, and f, pare Proposition 2. 3. restriction isomorphism
q u t(u) : Tq(E(u)) ...•T -r (E(u), E(u)) is the
#
beddings, then E(f ~) is the transverse
intersection of
i* : Tq(E(u)) ...•Tq(K) composed with the Thorn 'in M. The proposition then implies that we can make g transverse
~~om.orphism.
if
The result follows from the alternative definition of the
EW with E(1])
g: W ...•E(~) is a map, then
to E(f#~) in E(~), or make g transverse
to E(1]) in M. The result in either case is the same. Proof.
em-
We return to
these ideas in §4.
cup product, and the fact that the Thom isomorphism is composition with t(u).
See diagram.
Proposition 2. 6.
Let p be ..!heprojection of a mock bundle 1].
Then p! (p*~ u 1;) = ~ LJ p! ~. E(i*(~)) ---~~~
1 ~)) K
E(~)
!
Proof. '1'_
~ E(u)
There is also the Euler class of u, e(u) = i*t(u) which can be thought of as the result of intersecting
26
K with itself in E(u). Both the
~ E(I;)
Consider the diagram:
- - -i-)0
PI;
••
E(p*~)
)0
E(1])
E(~)
Jp~
-P
.•...• K
27
Triangulate
E(~), E(1/), K so blocks are subcomplexes and p~, pare
simplicial.
Assume p ~ = p ~
I
f
where K'
The result now follows from definitions and 2. 2. 3.
I 0
W
= 7T
f 1M defines an element of l' (X - U, A - U), and I
:
I
Mx I
=Mx
{o
n
= W ..• X
defines a bordism between
= MI
J, WI
o and let e: = d(M - int(A }, cl (U
CAP PRODUCTS AND DUALITY
<
e:j2. Define P
let M
Let X be a topological space, and define l' (X) to be the set of n (M, fl. Here M is a closed pl n-manifold,
= 1
=
I
I
argument applied to a bordism.
OIl
= f.,1 i = 1
1
and let
I
U
W
2
when Wand
Ware
0
I
restriction.
disjoint, and f(W ) 2
a : l' (X, A)'" T n
n
The following theorem is well-known.
n-
C
I
M with mesh The required
Injectivity follows by a similar
pl n-manifold,
Form f*[ ~]/M, and choose a repre-
1/. Then by 1. 2, E(1/} is a closed (n+q}-manifold,
is a map g: E (1/}'" I K I by composing.
and there
Notice that if f is simplicial,
#
then we can take 1/ = f (~).
f : (M, aM)- (X, A) with the notion of bordism enlarged to allow uW
are then easily checked.
~q/K be a mock bundle.
sentative
n
There is a homomorphism
and triangulate
n Let f : M ..• K be a map, where M is a closed
1
0, 1.
There are relative groups l' (X, A) defined by considering maps
= wo
where
The cap product
T (X) becomes a group by 'disjoint union' and is the nth pl n bordism group of X.
aw
1
I
f : M ..• X a continuous map, and (M , f ) is bordant to (M , f ) if n+ 1 0 0 I I there exists f: W ..• X, where W is a pl manifold with boundary g.: M. -W.
n,
f and f
M , give M a metric,
2nd derived neighbourhood of P (cf. [14]).
properties
the disjoint union W u W , and there are homeomorphisms
To construct
u of closed simplexes which meet M - A , and
bordism classes of pairs
such that fog.
J.
x {I
A.
l(A) given by
A sketch proof is
included, designed to generalise in Chapters ill, IV. Theorem 3. 1.
{T ( , ); a J n
n
is a homology theory on the category
of topological pairs. Proof.
Given a map f : (X, A)'" (Y, B), we get
n
n
Naturality of f* and a are
Exactness is proved by easy geometric arguments.
Hom-
otopy follows from the fact that M x I is an (n+l)-manifold with boundary M x {o J
U
We define [f] n [~] = [g], and the reader may check that we have a well-defined cap product
f* : T (X, A) ..• T (Y, B) by composition. then obvious.
.
M x {I J u aM x I. It remains to prove excision.
U C A with cl (U) C int(A).
Let
Remarks.
1.
There are obvious relative versions of the cap
products. 2.
The similarity
of the cup and cap products (using the second
description of the cup product) is clear.
Then we will show that
This will become more trans-
parent later when they are seen to be dual. i* : T (X - U, A - U) ..• l' (X, A) n n
Slant products
is an isomorphism.
1.
To see surjectivity, UI
=
I
f- (U), A
I
consider
= C I (A).
Then we have cl (U ) I
that there is a manifold M
I
28
f: (M, aM}'" (X, A), and define
C
M, with M - A
I
C
C
intM(A). I
M
I
C
Tq(X} ® TS(X x y} ..• Tq+s(y} [f]
®
[~]
•.•
7T
0
(f x ly)*[ ~].
We claim
M - U.
I
Then
29
f
x
Consider M x Y,~
that f: W -+ M is a map where W is an (n+q+l)-manifold
ly
Xx Y ~.K-+ Y. Then we have the bundle
•
(f
x ly)"~;M x Y, which we can regard as a bundle over Y by com-
posing with
the disjoint union f :W ....•MxI 1
11,
f
Tq(X) ® T (X x y) s [~]
iZl
[f]
We have to show that
U E(1)).
-+
T + (y) q s
1-+
Tly o([t]
sothat
1
r~x ly)).
~ x ly
f by a map Subdivide 1)
and the
Now consider the mock
bundle f : E(~) - J where J is the cell complex which has for cells 1
as a bundle over X x Y, take its
a
E
K x (1
U
0) of simplexes
a
a E
a.
E
M x I and the duals in each cell That ~ is a mock bundle follows
I
cap product with f, and then compose into Y. Remark.
in Tq(K).
1
cells of K x 0 U K x 1 are all subcomplexes. n
1)
and so that the blocks of ~ and
the duals in M x I to simplexes In other words, take
~-
and f-1(MXl)=1).
f-l(MXO)=~
so that it is simplicial,
1
2.
EW
Using collars on the boundaries of W, we can replace
11
with boundary
from [3]. Notice that ~ (K x 0)' is a subdivision of ~, and similarly
As before, there are relative versions which give in
particular slant products when X is replaced by (X, xo)' and X x Y by X A Y'" (X x Y, X v Y).
Y by (Y, Yo)'
for ~ I (K x 1)'.
Now subdivide ~ so that cells
of its base, for a
E
a x I are subcomplexes
K. Finally, amalgamate over K x I to realise the
required cobordism. Remark.
Poincare duality
The proof of duality shows that there is really no dis-
tinction between bordism and cobordism classes when the base is a maniLet M be a closed n-manifold and ~;M a q-mock bundle over some complex underlying M. Now the fundamental (bordism) class
[M]
of M is just the identity map 1: M -+ M, and since 1" = 1, [M] n ~ can be interpreted as the same map p ~ : E(~) ....•M, but regarded as a bordism
fold; they are represented by precisely the same class of map! The duality between cap and cup products is also clear from the definitions using the duality map; i. e., If.'W n 1) = If.'(~ U 1)), etc.
See the next
section for connections with transversatility.
class by 1. 2. In other words, we have the Poincare duality map General duality theorem Now let Y c X yC
defined by If.' {p : E - M} = {p : E ....•M }!
C
M be compact subpolyhedra,
for their complements.
We can regard
Xc, yC
c and denote X , also as compact
polyhedra by removing the interior of derived neighbourhoods of X and Theorem 3.2. Proof.
If.' is an isomorphism.
If.' is onto: Let f : W - M be a bordism class.
to find a mock bundle which 'amalgamates' simplicial,
Y. We have
to W. We can suppose f is
and consider the dual cell complex M* to M. Then
f : W - M* is the projection of a mock bundle. of Cohen's [3; 5.6]. a* equal to D(a,
Notice that for a*
E
This is a consequence
M*, we have the block over
0, which is a manifold with boundary corresponding to
the boundary of a*. If.'
30
is 1 : 1. Suppose ~q,
1]
q/X,
IK I =
M are mock bundles, and
Duality Theorem 3.3. q _ ~ : T (X, Y) T
Remarks.
+
n q
There is a natural isomorphism c
c
(Y , X ).
1. l/> can be regarded as an extension of the cap
product with [M]. 2.
(3.3) generalises both Lefshetz duality and Spanier-Whitehead
duality; e. g. , for the latter, take X = M = Sn (d. Whitehead [13]).
31
q By Spanier-Whitehead duality, T ( ,) is indeed the dual
3. theory to pi
submanifold.
Then we extend to give relative theorems,
embeddings and for general subpolyhedra.
bordism.
with block transversality Proof of 3.3.
Let N(X), N(Y) be derived neighbourhoods of
X and Y, and define
Ifi
to be the composition
I.
n
i*
We also give the connection
[12].
Let f : W ..• M be a map between compact pi closed, and suppose that N C M is a submanifold.
Tq(X, Y) ;:. Tq(N(X), N(Y)) ~ T +q(N(X) - N(Y), N(X) - N(Y))
theorems for
manifolds with W
Then we can regard
f : W ..• M* as the projection of a mock bundle ~, as in §3. Now ~ can be subdivided so that N is a subcomplex of the base (this involves a 1
homotopy of f); then C (N)
1*
is the restriction
manifold by 1. 2, and f is now transverse
of ~ to N, and hence a
(in some sense) to N!!
Notice that the proof is easily adapted to give an E-version by where a is amalgamation;
making the diameters of cells of M* < E. Further,
see Fig. 3.
by a whole family of manifolds. is a general subpolyhedron. verslity in this setting.
N can be replaced
In fact, the natural setting is where
Let X C M be a subpolyhedron.
f : W ..• M is mock transverse
In X, or f In
We say that
to X if f is the projection of a mock
bundle in which X underlies some subcomplex of the base. W
N
We now show how to treat relative trans-
We write
X.
= X n aM to a get a relative theorem (there are counterexamples otherwise; see 4.2 For technical reasons,
we need a condition on X
and 4. 3 below). We say that X that at each point x
E
X
is locally collared in (M, X) provided a ------there is a neighbourhood in (M, X) which is
a the product of a neighbourhood in (aM, X ) with the unit interval. a collaring is equivalent to collaring [8; p. 321].
Local
Fig. 3 'To see that
Let M be a compact mani-
and suppose
Then restrict
f : M ..• y
theorem 4. 1.
by Cohen's theorem. Ifi
regard
Relative transversality
fold with boundary and X C M a polyhedron with X = X n aM locally ---------a -1 collared in (M, X). Let f: W - M be a map such that f aM = oW,
To see
is surjective,
C
as the projection c of a mock bundle (in which the blocks might have extra boundary over X) Ifi
to X to get a genuine mock bundle.
is injective, combine this proof with the second half of the
flow
In Xo;
making f mock transverse
then there is an E-homotopy of f reI oW to X.
proof of 3. 2. Proof. 4. APPLICATION TO TRANSVERSALITY We observe that the mock bundle subdivision theorem (together with Cohen [3; 5. 6]) implies various transversality with the simplest case first,
Suppose f low
is the projection of the mock bundle UK,
and choose a ball complex L with
theorems.
We deal
the case of making a map transverse
to a
IL I = M
extending K, and so that X
is a subcomplex of L. This is done by first extending to a collar via the product ball complex K x I, and then choosing any suitable ball structure on M -
IK I x
[0, 1) and adjoining the two.
Following the proof of 3. 2,
we can suppose that f is the projection of a mock bundle ~ such that 32
33
~ 10M
is a subdivision of ~. Choose a further subdivision
so that L' <J L. with
~1 I K 3;~.
Theorem 4. 4.
~';1.' of ~
X
1
W if and only if W
In x.
Then amalgamating over L, we have ~ , say, over L 1
Thus Theorem 4. 1 (the version for embeddings) recovers
It only remains to observe that the homotopy of p ~
takes place within cells of K, and hence can be shrunk to the identity,
strengthened form of [12; 1. 2].
and this extends to give a modified homotopy of f by the HEP. Remark 4.2.
Proof.
In fact, the proof of 4. 1 used only that K extends
to L with X a subcomplex of L. on Xa than local collaring.
This needs a much weaker condition
A necessary and sufficient condition is that
the ambient intrinsic dimension [1] of X at X
is constant on the ina teriors of balls of K. This is always true if the ambient intrinsic dimension of X at x
E
Suppose X
II/W
so that X n E(v)
with
IK I = M
E(v)
1- W;
then there is a normal block bundle
is a union of blocks.
Choose a ball complex K
so that the blocks of v are balls of K, and so that X
underlies a subcomplex
(1.e., simply triangulate the complement of
and throw in the blocks of v!).
we
Then the inclusion
IKI
is
the projection of an (embedded) mock bundle with X a subcomplex of I the base.
Xa equals that of Xa at x.
a
Notice that the restriction
of this mock bundle to
gives
E(v)
the Thorn class of v. Example 4. 3. K n X is a I-cell.
Let X = 'the letter
Then, if f:
point of K n X, there is no homotopy of f reI aw making f mock trans-
suppose Wjp X by ~/K; then we construct a normal
block bundle v on W in M by induction over the skeleta of K so that it restricts
to a normal bundle for
~(a) in
(J
for each
(J
E
K. This
is an easy consequence of the relative existence theorem for block
verse to X. Transversality
Conversely,
T' with the top in aM so that
oW ....• aM is not transverse to the mid-
bundles [11; 4. 3]. Then X
for embeddings
Now suppose that f: W ....•M is a locally flat embedding (the condition on local-flatness
will be removed later).
E
locally-flat embedding (1.e., f-1(a) inclusion R~_C R:
K, we have f I
= aHa),
:
v.
Extension to polyhedra This subsection anticipates
We say that
f : W ...•.M is an embedded mock bundle if f is the projection of a mock bundle ~/K, and for each ball a
-.h
~(a) ...•.a is a proper
and f looks locally like the
for some k, n). We then observe that the sub-
gularities).
§3 of Chapter III (manifolds with sin-
Observe that if f: Y ....•M is a map where Y is a polyhedron,
and we apply the process of Cohen's theorem [3; 5.6] and regard
f as
a 'bundle' over M*, then Cohen's proof shows that the blocks, although not manifolds, are polyhedra with collarable 'boundaries'; M*, then f
-1 •
(a) is collarable
in f
-1
(a).
i. e., if
So we define f : y ...•. K to
division theorem for mock bundles applies to embedded mock bundles to
(J E
yield an embedded mock bundle, and that the homotopy which takes place
be a polyhedral mock bundle if for each
in the proof can be replaced by an ambient isotopy (by uniqueness of
in C1{(J).
collars [5]). Thus Theorem 4. 1 applies to give a relative transverality
versality theorem for two polyhedra in a manifold and similar relative
theorem for embeddings via an E-ambient isotopy (fixed on aM).
versions and versions for embeddings.
E
K we have f
-1 • ((J)
collarable
Then the subdivision theorem works and we thus get a trans-
lransversality
Connection with block transversality
(J
implies transversality
In the case of embeddings,
mock
in the sense of Armstrong [2].
This is proved by using the collars to construct neighbourhoods of the Suppose X C M is a compact polyhedron and W is a locally flat submanifold of M. Recall [12] that X is block transverse
to W in M
if there is a normal block bundle v /W in M such that X n E{V) E(vIX
34
n W). We write
X
-.h
W or
xlv.
=
form cone x transverse
star;
compare with the proof of 1. 2. McCrory
[15]has shown that mock transversality equivalent to both block transversality
for polyhedra is symmetric
and
in the sense of Stone [16] and to
35
i
transimpliciality
in the sense of Armstrong
The transversality
definition
Suppose given
I inn {}0 ...• An+l. gIven by r:.. x II A x
[2].
of the cup product
~q, r{;K
then for some large
m we may assume
(t , .•. , t , s) ~ o non
«1 - s)t , ... , (1 - s)t , s).
m This determines a natural isomorphism E W, E (1]) elK I x 1 so that PeP are restrictions of the projection m IKI x 1 ...• IKI. Now consider E(~)1]X1m E(1]) x 1m c /KI x 1m x 1m also based homotopy equivalences .. m 0' 1 0 l' By mductIvely making ~(a) x I transverse to 17(a) x 1m in ax 1m x lID o 1 0 1 I/J(Y): ISY/ ...•Y and 1/I(X):x we get a mock bundle E(~) x I~ n E(1]) x I~ ..• K It can easily be seen that this gives the cup product,
using the alternative
definition.
we sketch a proof below connecting the transversality restriction
to the diagonal definition.
generalising
s2m(~ x 1]) =
denote q-fold suspension.
considered
(see [10; p. 334]). l/J(nlxl)
This proof has the virtue of
to the more complex situation
Let s
However
Consider
definition with the lelxll
0
e(Y) : nsy
...•sny.
There are
...•slxl
the composition
° In1/l(x) I
:
Inxl
..• lnslxll
...•lsnlxll
...•nlxl.
in Chapter V. We have now proved:
From definitions
q m m smW x t(K x I ) u t(K x I ) x sm(1]) and the total spaces
on the right are transverse without being moved. Let i: IK I x 12m ...• 2m IKI x IKI x 1 be given by i(x, y) = (x, x, y). Then applying i* we get
Lemma 5. 1.
Suppose
X is a Kan based A-set.
There
is a based
(weak) homotoEY eq~ival~nce Inxl-nlxl. Now define an n-spectrum a Kan A-set
Desuspending
both sides reveals
the coincidence
S
of definitions. em :
5. THE CLASSIFYING SPECTRUM
For each
and A-sets
the category
We give a simple- minded definition A-sets.
Basic facts about A-sets
Given a based Kan A-set
contained
of spectra
in the category
of pairs
nx
i "
=
of S (PL)
0 1, ... , n, are just restrictions
of the
for
cobordism
pl
S (PL)
as follows. m
Let R co = URn.
Then a k-simplex
kook
denote the space of loops on the based complex of Y.
CW complex
Y,
There is an identification
Base simplexes face operators
*k
€
S(PL)m
are defined by taking
are defined by restriction.
1. 1 and general position that
36
of maps by
projection.
I
and let SY denote the singular
and homotopy classes
is a compact polyhedron X c A x R such that 71 I : X - A m k is the projection of an m-mock bundle over A , where 71 is the natural
o~+1a = * o' The operators
0. in X,
Let ny
h* is defined on
as follows, Define
I
of CW complexes
theory
of
The n-spectra
O. : (nX)(n) ..• (nX)(n-l)
Z is given
in [10] will be assumed.
X, we define a based Kan A-set
if and only if 0n+l a = *nand
€
-ns m- l'
It follows from 5. 1 and [13] that a cohomology
n-spectra
m
and a homotopy equivalence
m
Sm
as follows.
S (PL)
m
X
=
¢, and
It follows from the proof of is a Kan complex (compare
[11;
37
k k k+ 1 k() 1 1 Dehne e : 6. -+ 6. byes = 2S + 2vk+l' ek : !j(PL)(k) -+ (Ug(PL) )(k) defined by o
2.3]). m
m-1
k
00
L\. x ReI
X c
e
m
gives a mock bundle 11 /CK such that s ~/K x I is isomorphic to the # amalgamation of the pull- back 1f 11 by the pinching map 1f: (Kx I)' - K,
Then we have
: g(PL)
m
where (K x I)'
x oct
A
- U!j(PL)
m-
Proposition 5. 2.
.
m
: g (PL)
m
-+
ug (PL)
m-
Theorem 5.3.
e
m
1 is a homotopy ------
[1]
is injective so we have to describe a deformation
retraction of ug (PL)
on e
m-l 'sliding to the half-way level'.
(g (PL) ). This can be thought of as m m More precisely, suppose given a map
0
0
n 1
n,l
0
0
a co~e on A with cone-point last. Then E(~) lies over the half-way [4] n,l level in C oA and we homotope E(~) reI boundary into the pre- image n, l' of the half-way level over CA o' Then, using an identification of [5] n,l n C6. with CA x I and general position, we get an n-simplex of 0
0
0
n, 1
whose restriction
th
to the i
face lies in em(g(PL)m)'
[6]
The
deformation then follows from [10; 6. 3].
[7]
Now let K be an ordered simplicial complex, and f : K-g (PL)m a 6.-map. Then we can form an m-mock bundle UK by gluing together empty bundle, we get a function
[8]
-----
{g (PL) m ,e m
1.
m
is a bijection,
86 (1967),
(K, L),
[10]
M. M. Cohen. Simplicial structures
for polyhedra.
Ann. of Math.
Ann. of Math. 85 (1967),
and transverse
cellularity.
218-45.
M. M. Cohen and D. P. Sullivan. On the regular neighbourhood of a two-sided submanifold.
Topology, 9 (1970),
J. F. P. Hudson and E. C. Zeeman. Publ. I. H. E. S., 19 (1964), H. Morton.
141-7.
On combinatorial isotopy.
69-94.
Joins of polyhedra.
Topology, 9 (1970),
F. S. Quinn. A geometric formulation of surgery. C. P. Rourke. C. P. Rourke.
243-9.
Ph. D. thesis,
1969.
Covering the track of an isotopy.
Proc. Amer.
320-4.
Block structures
in geometric and algebraic
C. P. Rourke and B. J. Sanderson. Oxford, 2, 22 (1971),
Then by general position (com-
and it follows from [10] that induces [11]
[12]
for polyhedral pairs. Finally, we notice that the suspension isomorphism essentially More precisely,
G. W. Whitehead 102 (1962),
Quart. J. Math.
Block bundles I.
Ann. of
1-2B.
C. P. Rourke and B. J. Sanderson. Math., 87 (1968),
[13]
6.-sets I.
321-8.
C. P. Rourke and B. J. Sanderson. Math., 87 (1968),
a natural bijection
coincides with the function em'
Transversality
172-91.
topology. Report to I. C. M., Nice, 1970.
where []6. denote Jl-homotopy classes. § 2]),
M. A. Armstrong.
Trans.
413-73.
Math. Soc., 18 (1967), [9]
: [K, L; g (PL)m' *]Jl - T
143 (1969),
Princeton University,
the images of simplexes of K, and since the base complex gives the
38
q
E. Akin. Manifold phenomena in the theory of polyhedra. A. M. S.,
[2]
A -+ ug (PL) whose boundary goes into e (g (PL) ). Then glue n 1 m-l m m the'simplexes together to form a mock bundle UCA embedded in [3] n,l CA x Roo and empty over cone-point and base, where CA denotes
pare [11;
{Tq, 0 } coincides for
REFERENCES FOR CHAPTER II
Proof.
m
The cohomology theory
-----~~-~
polyhedral pairs with the cohomology theory defined by "--"-----=--"c=..-_~~ ~
equivalence.
Ug(PL)
is simplicial.
We have proved:
1 is then a based 6.-map. e
1f
00
XR •••..• k+ 1
Block bundles
n.
Ann. of
256- 78.
Generalised homology theories.
Trans.
A. M. S.
227-83.
em 0 f: K-U(g(PL)m_ll 39
E. C. Zeeman.
[14]
Seminar on combinatorial topology. 1.H. E. S.
and Warwick, 1963-66. C. McCrory.
[15]
Cone complexes and pl
transversality.
III· Coefficients
Trans.
A. M. S. 207 (1975), 1-23.
D. Stone. Stratified polyhedra.
[16]
Springer-Verlag lecture notes,
no. 252. In this chapter we give a geometric treatment of coefficients in oriented pl
bordism theory.
The definition (although not all the
theorems) extend to other geometric homology theories and this extension will be covered in Chapter IV, as will the extension to cobordism (mock bundle) theories.
Application to general homology theories and con-
nection with other definitions of coefficients will be covered in Chapter VII.
There are two good definitions of coefficients: 1.
For a short resolution p of an abelian group G we define
coefficients in p by labelling with generators and introducing one stratum of singularities of codimension 1 corresponding to the relations (see §1). 2.
We allow labelling by any group elements,
corresponding to any relation and then, in the bordisms,
and singularities allow singulari-
ties of codimension 2 corresponding to 'relations between relations
I
(see §3). Definition 1 is very simple geometrically while definition 2 is functorial in G. To prove equivalence of the two definitions involves a further definition, for longer resolutions (in §2). The basic geometrical trick is resolution of singularities and appears in the proof of the universal coefficient sequence in §2. The universal coefficient sequence itself can be seen as the measure of the obstruction to resolution of the final singularity.
In §3 it is seen that the universal coefficient sequence is natural
for G; consequently by [3] it splits for a large class of abelian groups, including all groups of finite type. In §4 we show how to regard the product of a (G, i)-manifold with a (G', i')-manifold as a (G®G', i+i')-manifold
and thus define a cross
product for bordism with coefficients.
40
41
In §S is the Bockstein sequence and in §6we observe that, if G is an R-module, then n*(-, -; G) inherits an U*(Pt.; R)-module
3. For each component (Q, r) of S(P) there is given a regular neighbourhood N of Q in P and an isomorphism
structure in a natural way. Using Dold [1] we then have a spectral sequence Tor (U (-, -; R), G) o-? U*(-, -; G). p q We use the convention - ~ + for inducing orientations on the boundary of I-manifolds,
and in general use the 'inward normal last'
h:
N ...• Q x C(L(r,
p))
where C(X) denotes the cone on X and h carries
Q by the identity
to Q x (cone point).
convention.
4. h is an isomorphism of F 0 -manifolds off Q. Intuitively, a p-manifold is a manifold labelled by generators for
1. COEFFICIENTS IN A SHORTRESOLUTION
G with codimension 1 singularity labelled by relations. Let F be a free abelian group with basis
B. We define co-
sheets and orientations of P near S(P) give the relation labelling the singularity.
efficients in F by labelling with elements of B, i. e. define an Fmanifold to be an (oriented pi) manifold each component of which is labelled by an element of B. We write (M, b) or Mi&> b for M labelled by b. It is easy to see that bordism of F-manifolds defines a bordism
Examples 1. 1. F 0 = F, Bo = B.
theory n*(,; F) and that U*(X, A; F) ~ U*(X, A) i&>Ffor the pair (X, A). Now let G be a general abelian group and
p
2.
a short resolution O-Z
of G. p comprises a short exact sequence 1/>1 1
...•G 0
and F are free abelian groups with bases Band B . o 1 0 1 The elements of Bo are the generators of G and those of B1 are the relations for G with respect to the resolution
p.
We will define co-
efficients in p by starting with U*(,; F ) and 'killing' the elements of o U (pt.; F ) which correspond to the relations:o 0 Let r E B then I/> (r) E F and can be written uniquely in the 1
L O'.b. where 111
b.
1
E
B.
0
1
0'.
The element
n
G = F a free abelian group and F
1
=
0 '
and we use the usual presentation
Xn ...• Z ...• Z
Then B
•..• O
where F
form
G= Z
1.
Then a p-manifold is precisely an F-manifold.
n
...• 0
= {Il, B1 o and can be suppressed. L(I,
E
•..• F
O •..• F
Moreover the
•
= p)
{Il
and so the labels give no information
is the union of n points (all oriented +).
Thus a p-manifold is a manifold with a codimension 1 singularity, has a trivial neighbourhood of the form C(n points).
which
Moreover the
orientations of the n sheets all induce the same orientation on the singulatiry.
This is precisely Sullivan's description of 'Zn-manifold'
[4].
See Fig. 4.
0
is an integer and the sum is taken over elements L(r,
labelled by b. and oriented 1
p)
E
U (pt. ; F ) is the union of 0
+ if
union is taken over all elements
0'. 1
b.
1
E
>
0
0 and - if
0'. 1
10' i I
points
< 0, where the
B . 0
A p-manifold of dimension n is a polyhedron P with two strata P =:J S(P), labellings and extra structure such that 1. P - S(P) is an F -manifold of dimension n. o 2. S(P) is an F -manifold of dimension (n - 1). 1
Fig. 4. Part of the neighbourhood of the singularity in a 'Z manifold' 3 42
43
3.
G= Z
00'
merge together.
We describe the resolution by specifying the
p
generators and relations.
The generators are 'liP',
2
'liP
"
•••
and
il the relations, as elements of F , are (P('l/pi,) - 'liP - ,), i= 1, 2 ... o . and p('liP'). Thus the sheets are labelled 'l/pl, and the singularities occur where p sheets labelled 'l/pi, 'l/pi-l,
tion has the obvious compatibility.
merge together.
notion of 'Z3 -manifold' in Example 2, as we will show in §3. There is a natural notion of p-manifold with boundary and we thus have a bordism theory n*(,;
p).
That this theory is a generalised
homology theory follows from the proof given in II 3.1 for T *( ,).
merge into one sheet labelled
or where p sheets labelled 'l/p'
This notion gives the same bordism theory as the
Orienta-
(The crucial fact is that the regular neighbourhood of a polyhedron in a p-manifold can be given an essentially unique structure
See Fig. 5, in which p = 5:
as a p-manifold )
We now turn to the universal coefficient theorem for p-bordism.
In §2
we will prove the theorem in the general case (for longer resolutions) and here we will content ourselves with the statement and a sketch proof of this (simpler) case, with stress on the geometry of the situation. Theorem 1. 2.
Let p be a short resolution.
There is a short
exact sequence, natural in (X, A): l
0'" n (X, A) ® G'" n (X, A; p) n
n
Sketch of proof.
relations
n-
1(X, A), G)'" O.
(For full proof see 2. 5.)
The notation is intended to suggest that l
G = Q, the rationals. p('l/n')
Tor(n
The spaces
(X, A)
play no role in the proof of the theorem and we will ignore them.
Fig. 5 4.
s -
- 'l/q',
Generators
'l/n',
n = 1, 2, .,.
and
where n = pq and p is the smallest prime
and s is 'restriction
is the 'labelling' map
to the singularity'.
Description of l
occurring in n. The picture is similar to the last one. Using the description of n 5.
In all the cases above we have chosen the most natural resolu-
tion for G. Other resolutions also give rise to a notion of 'G-manifold'.
ments of B 0 II : nn ® F 0
® F as manifolds labelled by elen 0 (see start of this section) we have a 'labelling' map
•••
nn(p)·
Now II
is zero on relations.
In the example below, a, b, c, d are the generators of the copies of Z
follows. Let r be a relation and [M] En.
indicated:
manifold M x L(r,
n
ZGlZ-ZGlZ--+-Z
p)
b-l
II ([M], r) to zero.
morphism l : n
Consider the labelled
x I with each copy of M x {O}
gether and this end labelled by r. 3
This is seen as identified to-
This constructs a p-bordism of
See Fig. 6. It follows that II
defines a mono-
® G'" n (p) n n'
c~a-2b dl
••
a+b
Here a p-manifold has two sorts of sheet: a-sheet and b-sheet. Two b-sheets can merge into an a-sheet and an a and b sheet can
44
45
W
1
where W
r
=
Mx I
U
W x C(L(r, r
p»
is a typical component of W (labelled by r) and the union
identifies the obvious subset of M x {l ) with aW
r
is a p-bordism of M to an F -manifold. a
x C(L(r, p)).
WI
See Fig. 7.
M
M x L(r, p)
M x I
Fig. 6
p»
W x C(L(r,
Description of s Let P be a p-manifold with singularity
S(P).
W
S(P) is an F 1-
manifold, moreover the map
Fig. 7
®lJll
n n-l ®F
1
-
n n- l®F, a
can be described on generators of S(P) in n
®F
as product with L(r, p).
is represented
n-l a neighbourhood of S(P) in P.
2. COEFFICIENTS
by
au,
. nn (P) - Tor(n n-l' s. argument.
Hence the image
where U is a regular
In this section we will generalise the construction coefficients in resolutions
This is bordant to zero in nn_l ® Fa
since it bounds cl (P - U). Thus the 'singularity'
IN A LONGER RESOLUTION
defines a map
G) and it is surjective by reversing the above
Let G be an abelian group. (a)
1Jl
A structured
resolution
p
of G
1Jl
2
1Jl
1
:5
4
E
O-F -F -F -F -G-0 3
Order 2 is trivial (an F -manifold has no singularity). To see a exactness, suppose M is a p-manifold with S(M) bordant to zero as an
46
of §l to give
4.
a free resolution of G of length 3
1
:5
consists of
Exactness at 0n(P)
F -manifold by a bordism
of length
W. Construct the bordism
2
1
o
BP, for each F (p = 0, 1, 2, 3) P (c) for each bP E BP (p = 1, 2, 3) we are given an unordered word, w(ll), representing the element IJlp(bP). Precisely w(ll) is a finite set (of pairs) (b)
a basis,
47
Construction of .cP such that ~ 1.0.b~-1 (bP), where the sum denotes the element of Fp - 1 1 1 = ""1'p obtained from w(bP) by adding up all the coefficients belonging to the l same element in BP- . For each bP
(d)
defined as follows.
E BP (p = 2, 3) we are given a 'cancellation rule' Let w(bP) = {Oibr 1 : i E I(bP) ). Then order two
of p implies that the formal sum Z;.0.w(b-?-1) is an unordered word 11 1 representing the zero element of Fp-2 in terms of the elements of BP-2• The effect of O. is an inversion of sign iff 01,= '-'. 1 The given cancellation rule consists of a procedure for pairing off the letters of 2;.o.w(b-?t -1) in F 2' Precisely c(bP) is a partition of p-
1 1
. 0_ 2
the letters of LiOiw(bf-1) into pairs of the form (Ojlr
• 0- 2
, 0klr
.
) w1th
O. '" Ok' J For the sake of simplicity we may write the set I(bP) in the form {I, ... , l : l
=
l (bP)
J.
If a ~ k ~ 3 then p
For diagrams we refer to the examples given later. 1
.cO let b
E
B\
k
1
(or generated by b the class
the orientation
1, ... , l)
0.. 1
1
1
in p) and will be written
L(b
1
1
1
The
in p
We define
pl.
,
1
with b
p)
,
1
E
1
B
,
i. e.
1
.cO =' {L(b , p) : b E B J. Now consider the join of L(b , p) and the 1 1 1 1 1 1 point b , written b L(b , p), and give b L(b , p) - b (= the open 1
cylinder over L(b '-'
sign).
,
p»
the orientation
1
1
The join b L(b
1
,
1
over L(b1, p) with vertex elements of B
1
'+, on b
with the orientation
+ (arrow departing from
- -+-
p), with the above orientation off b
,
1
b
will be referred
and
to as the (oriented) cone
In general it happens that different
•
may give rise to the same a-link.
Therefore,
vertices generating the link. To construct
Clearly Po
=
p.
(G, p, p, n), where G is an abelian group,
0, we shall construct
(a) a class of pl isomorphism types of polyhedra with extra called the class of (p, p, n)-manifolds. .cP of (p, p-I, p)-manifolds for p
2:
0, called the
class of (p, p)-links. We start by defining a (p, -1, n)-manifold to be an F o-manifold and in general (p, p, n)-manifolds will be defined from (p, p-I, n)manifolds by 'killing' the class .cp. The precise definition (an extension p of that in §I) is given at the end of the construction of the classes .c . The (p, 2, n)-manifolds will be called simply (p, n)-manifolds.
over the
:
2
2
.c , let b
2 E
2
B ; w(b )
=
p ) as in the previous case. 1
l (b
)
1
2: o.b .. Construct i=I 1 1 Each b~ generates a 1
Q-link L(b~, p) (i = 1, ... , l); consider the space ~ o.[b~L(b~, p)] = L 1 i=11 1 1 where b~L(b~, p) is the cone as defined above and O. changes all the 1 1 1 orientations present in this cone iff 6. = '-'. Let Vi = ~ 0.w(b~) = 1 1 1 lbO, ... , b~ with the appropriate signs L Then L = ~ o.L(b~, p) is 1 i=I1 1 obtained by giving Vi the discrete topology and orientations according to the signs.
Therefore the cancellation rule c(b2)
gives a canonical
way of joining the points of L in pairs by plugging in oriented I-disks. Precisely suppose 0jb~ is paired with Ok1\.. Then OJ'" Ok and we insert a I-disk
[b;, ~]
'arrow departing from
=
unique element b;
b~
with orientation given according to the rule
,+, E
sign'. Moreover we label the I-disk by the O B • The object which is obtained from L
through the above identifications in L is called the I-link associated to 2
b
2
in p (or generated by b2 in p) and it is written
class of I-links,
48
=
.cO to consist of all polyhedra L(b
1
p is a structured resolution of G; p, n are integers such that
(b) a class
{b~, ... , b~} the dis-
same link, there may be different cones, corresponding to different is the structured resolution of 1m ¢k
where the structure is that induced from p.
structure,
Give the set
1 1
resulting polyhedron will be called the a-link associated to b
the a-link L(b2
2:
Z; o.b? i=I
1
•.•. Fk •.•. 1m ¢k .•.•a
-1 ~ P ~ 2; n
=
)
crete topology and each point b. (i
t/>k
For each quadruple
1
w(b
To construct
l
.c
1
,
is defined by
£1
=' {L(b
2
,
L(b
p) : b
2
E
pl.
,
B
2
The
}.
49
It is clear that it is the given cancellation rule in p that makes the construction well defined.
W is an oriented manifold,· are defined as in the previous cases.
Different rules may give rise to com-
We now define (p, p, n)-manifolds (without boundary) inductively
pletely different links.
on p as follows.
We think of L(b2, the intrinsic j-stratum
as a one-dimensional stratified set in which
p)
2
is the O-link generated by b
in
p . I
2
It is clear how to define the cone b2 L(b , p): 2
2
.
2
b L(b , p) IS the usual cone over L(b , p) wIth vertex b , the subcone 2
over the O-stratum of L(b ,
2
is given an orientation outside b
p)
S(M)
(a) M - S(M) is a (p, p-l, (b) S(M)
n)-manifold
is an FpH-manifold
(c) (Trivialised stratification condition) For each component V l of S(M) labelled by bP+ E BP+\ there is given a regular neighbourhood
topologically
2.
M:)
with labellings and extra structure such that
=
0, 1) consists of a disjoint union of j-disks, lj each disk is oriented and labelled by one element of B - ; the O-stratum (j
A (p, p, n)-manifold is a polyhedral pair
N of V in M and an isomorphism
as
in the previous step; the subcone over the I-stratum has the orientation given by the cartesian product: (I-stratum) 'from
- to +'.
(d) h is an isomorphism of (p, p-l, 2
Finally the vertex
2
2
stratum of b L(b ,
b
has the orientation
+.
Each
Remarks 2.1.
is labelled in the obvious way. Now, as before,
p)
2
it may well happen that different elements in B
generate the same
2.
2
topological product W x b L(b ,
From now on we think of the above
p).
product as having the following additional structure:
,
W xL,
stratum of L(b2, p)
(j
W xL,
°
system with base L(b
p
the product orientation on each stratum.
I
3
=
3
2. Let b E B
•
Consider
striction of the normal bundle to aD is aD x L(b\ 1
extend the bundle by plugging in D x L(b of L(b
,
,
) 2
50
2
w(b
labelled by elem.ents of BO. Then plug in an
p)
3
3
and the product W x b L(b
,
p),
where
) 1
w(b
1
,
pl.
3
E Ker ¢
=
_b
=
1
b
2
1
-
¢1
F Ker e: -
FZ 3 - Z 3 - 0
3
1
+ b
11 _
21
=_b
bi
12
22
31
+b
=
bO + bO
w(b1 )
=
-b
1
E F Ker e:
+ b1
1
)
11
12
2
2
1
1
) 3
w(b1
¢2
F Ker ¢
-
be such that
1
2
3
The cone b3L(b3,
w(b
I/>
= _b2 + b2 + b2
2
pl. As a result of clothing
oriented labelled 2-disk for each sphere and get the required link L(b
1/>3
p : 0 - Ker
w(b3)
P ) we are left with a polyhedron, whose boun-
dary consists of I-spheres,
Then the notions of p-manifold defined in §l and §2 coincide.
Let b
therefore we can
p);
A short resolution gives rise to an obvious
structured resolution (the cancellation rule gives no information in this
2.
as follows. If 6.b~ is a vertex of III L(b , P ) then put 6.L(b~, p) as the fibre at that vertex. The part of III 3 L(b , P ) which remains unclothed consists of a disjoint union of closed I i I I-disks. Let D, labelled by b E B , be one of such disks. The re-
the I-stratum
1.
and construct a trivial normal bundle
3
3
Examples 2.2. case).
p )
,
The definition extends to yield (p, n)-manifolds with boun-
dary in the obvious way.
J
We are now left with the case p the I-link associated to b3 in
p has order two.
3.
0, 1); a labelling on each stratum obtained from
the labelling of the second factor;
3
three intrinsic strata,
p)-manifold (and this completes
L. being the intrinsic j-dimensional I
=
It is obvious how to give a p-link the required
In all the above definitions we have only used the fact that
the resolution
Now suppose W is an oriented manifold, then we can form the 2
1.
extra structure to make it into a (p, p-l, .all the definitions).
I-link and therefore over the same link there may be different cones.
namely W x b2
n)-manifolds off V.
x [-; +) where [-; +) is the half open I-disk oriented
23 1
32
+b
33
E F Ker e: EFKere:
2
° - b°
1
2
51
w(b1
)
'(b1
)
w(b1
)
21
\\
= bO + bO 1
22
°
1 33
)
1
CP2
° °
1
= -b 12
Let b
12
21
w(b2)
22
° - b°
1
W(b
2
=bO EFZ . bO =bO 1 3
2
2
F Ker cP
-
1
CPl
F Ker e - FZ
-
be a basis element of F Ker
5
- Z - O. 5
Suppose
CPl'
2
11 1
P : 0 - Ker cp
3.
2
3'
21
E(bO)= 1 EZ ; E(bo) = 2 EZ 1
22
= bO + bO + bO + bO + bO
bO =bO 11
21
= bO + bO + bO
32
w(b
12
=bO +bO +bO
23
w(b1 ) 31 w(b1 )
11
rules
being suggested by the diagram itself.
= bO + bO + bO + bO + bO
°
of L(b3, p), the cancellation
Figure 8 shows the construction
2
2
22 3
=bO EFZ . E(bO)= 0 EZ . 2
3'
°
3'
) 1
1
1
1
1
2
3
= b + b + b + b1 = bO + bO + bO 11
12
w(b1) = _be 2
4
13
+ bO + bO
21
22
23
w(b1) = _be _ bO + bO 3
w(b1)
31
32
33
= _be _ bO _ bO
4
41
°
43
E(b11) = E(b
°
42 )
43
°
= 2; e:(b
°
= e:(b
)
1
)
4
= 3;
e(bo ) = e:(bo ) = e(bo ) = E(bo ) = O· 13
21
e(b 0)
= E(b0)
22
32
33
= 1;
e:(b°
41
)
23
'
= E(b0)
31
= 4.
Fig. 9 Figure 9 shows two possible links associated Fig. 8
ponding to different cancellation rules.
to b2
This completes
in p corres-
the examples.
Now let M be a (p, n)-manifold and M' a (p, n')-manifold. embedding f : M' - M is a locally flat stratified
52
An
embedding between the
53
underlying polyhedra, trivialisations.
which is compatible with the labelling and the
If n' = n, then f may be orientation preserving
orientation-reversing.
In the following, unless otherwise stated,
Proof.
First of all we anticipate that the proof consists
or
geometrical arguments involving only (p, n)-manifolds
a co-
fications.
In the constructions,
dimension zero embedding will always be assumed to be orientation preserving.
(X, A) = (point, ¢).
-----
M is a subset
M
°C
M
together with an embedding f : M' '+ M (of p-manifolds) such that f(M') = M.
°
obtained from
If M is a (p, n)-manifold, M by reversing
(p, n)-manifolds
have the following properties. Proposition natural structure
2.3.
1.
of M with I, it is clear that 2.
It involves a resolution of singularities. (p, n)-manifold.
!! M is a (p, n)-manifold, M x I has a
where
>
SM has codimension
p.
M is bordant to a (p, n)-manifold
Q, whose last stratum
°
The proof of 2. 3 is left to the reader.
There is an obvious notion
of a singular (p, n)-manifold in a space and thus we have the bordism group n (X, A; n
pl. The following proposition follows directly from
proposition 2. 3, using the proof of II 3. 1. Proposition
2.4.
{nn ( , ; p)}
on the category of topological spaces.
2. 5.
SM is
Then
is S, by a
p.
-SM and -S:
assume that SR consists of a set of equally labelled components, with label, say, bP E BP; v(-SM) SR
x
=
normal bundle of -8M in -M=MX
pl.
L(~,
v(-SM)
{-I};
C
M x {-I};
v(-SM)
C
SR x L(~,
pl.
So we can form
theidentification space: =is=--=a~g",:e=-=n=-=e:..:.r--=a:..:.li=s--=e:..:.d--=h-=-o:..:m=-=o-=-lo::."gy,,-,,-..:.:th-=-e:..:o:..::r.:!-y R = SR x L(bP, p) ~)) M X I'
For each integer
there exists a short exact sequence
SR = any bordism between
We have:
The universal coefficient sequence Proposition
Let SM be
We need the following:
submanifolds of OM, aM' res_.p~e:..:c:..:t-=-iv:..:e:..:l,,-y--=s::..:u::.:c:..:.h=--t.:...h.:...a_t M 0 ~ -M', then _______ bordism R whose last stratum is still in codimension MUM' is a (p, n)-manifold with boundary isomorphic to g ---"__ --'0--__ Consider the following spaces: Proof. Cl(aM\M U aM'\M'). o g 0 M x 1', where l' = [0, -1]; 3. Let M be a (p, n)-manifold and X C M. Let N be a regular neighbourhood of X in M. Then N can be given the structure -~~ (p, n)-manifold in an essentially unique way.
1.
Suppose that [SM] = [8] as Fp -manifolds.
Lemma 2. 6.
If M, M' -are (p, n)-manifolds and M 0gO-, M' are (p, n-l)_
in codim
the last stratum of M. Then, by definition of (p, n)-manifold,
obtained by crossing the structure .anFp-manifold,
a(M x I) ~ M u - M u aM x 1.
Let M be a closed
We are going to show that there exists a (p, n)-manifold
M, bordant to M and having no singularities
of (p, n+ I)-manifold,
we shall assume
Description of the map s
-M denotes the (p, n)-manifold
all the orientations;
and their strati-
the maps into (X, A) do not play an
essential role and so, for the sake of simplicity,
A submanifold of a (p, n)-manifold
of
hich provides the required bordism. n
2:
0 and each pair
(X, A)
If SR has many labelling elements, truction simultaneously
we perform
the above con-
on every set of equally labelled components.
rhe last stratum of R is SR and has codimension
p.
See Fig. 10.
which is natural in (X, A).
54
55
is exact and so there exists an element
because if SM is only bordant to 8W I8iw(~+l):
8M - SW I8iw(~+l),
then, by Lemma 2. 6, M can be replaced by another
(a)
M such that:
M is bordant to M by means of a bordism with singularities
up to codimension p only (b)
SM = SW I8iw(bP+I),
Therefore assume
8M = SW I8iw(bP+I),
If SW - ¢ we are reduced to the
case of Remark 3 and we know how to solve the singularities
II'
assume
then,
So
SW '" ¢ and take the following spaces: M x I, where 1=[0, -1] -(SW x bP+lL(bP+l : p))
Fig. 10 Remark 2. 7.
If we can choose S = ¢ then the above constructio
gives a resolution of the low dimensional singularities words, when the low dimensional singularities having the same kind of singularities
v(-SM)
= normal bundle system of -8M in M x {-I} = -M.
of M. In other
of a (p, n)-manifold
M
bound (in a labelled sense), they can be resolved by means of a bordism
Then we have v(-SM)
C
M x I and v(-SM)
C
-(SW x bP+1L(bP+1, p).
The identification space
as M. (Cf. proof of exactness in W=_(SWXbP+lL(bP+l,
1. 2. )
p)
'-------"
MXI
v(-S(M)
Proof of Proposition 2. 5 (continued).
Now let us look at the
image of [SM] through the morphism _ I/> : P
nn-p I8iF p
id I8iI/>p nn-p I8iF p-l . i
i
C
of Jt~e~oundar~ ~f ~he cdmPlement of a regular neighbourhood of V
in the (n-p+ I)-stratum
and a (p, n)-manifold
n - p + 1. The singularities
M' 8M have
been resolved up to bardism. In general, if SW is of the form nP+ 1 SW= ~k(SW)k I8iK ' then one performs the above construction simul-
SM ~ ([V.] I8ib.) = [V.] I8iw(llltaneously on all terms 'pI I 11K .6.b~-1 = L .[6.V.] I8i})\J-I: this is nothing else than the bordism See Fig, 11.
,
&> ~
realises a bordism between M = M x {D} whose last stratum has dimension
We have for each component V I8iIf cl~s
I8iF
8M = SW I8iw(~+l), (p, n)-manifold
The whole manifold is R
[V.]
n n-p
such that p+l ~P+l[SW] = [SM], Suppose first that SW is a set of components all labelled by bP+1 € BP+1, We can always reduce to the case
M x II
[SW] €
(SW)kI8iiJ'+I
and gets the desired manifold M'.
We remark that in order to get rid of singularities
of M. Therefore the image of [8M] is the bar- p we have used bordisms,
which have singularities
in codimension
up to codimension
dism class of the boundary of the complement of a regular neighbourhood P + 1. SM in the (n-p+ I)-stratum
n
n-p
I8iF
p-
I'
of M and, as such, is the zero element of
Now, because ~p+l
p>
1, the sequence: I/>p
nn-p I8iFp+l - nn-p I8iFp - nn-p I8iFp-l
56
So now we have a well defined procedure to solve the singularities of a (p, n)-manifold
M, stratum by stratum,
starting from the last one
and going up by one dimension each time, until we are left with a (p, n)manifold,
lVi,
which is bordant to M and has singularities
dimension one at most.
In general we cannot solve
8M in co-
SM as above, because
57
Proof.
a closed polyhedron, because the singularities p at most.
p + 1 is
The intrinsic stratum of M in codimension
oM are in codimension
of
Therefore the above construction for solving the singularities
can be applied, essentially unaltered,
p + 2
to solve the codimension
oM by a bordism with boundary, W, OW. No new singu-
stratum of M,
larities are created along the boundary during this process. I
-~
-----M'
Proof of Proposition 2.5
(continued).
Each M
E
[M]
-
deter-
p
mines an element of Ker cp , namely [8M]. This assignment depends on I
/
M; in fact, if MI. - M, then s1\lt I may be bordant to
the representative _
8M by means of a bordism, V, with singularities [8MI]
'" [8M].
and therefore in general
However, by Lemma 2. 8, the singularities
of V can be
assumed to have codimension one at most and we can certainly write: [8M] = [8MI] + [N] where [N] The whole manifold is W.
H (p,
-
I
[M] ~
n n- 1®F
2
2 -
~I [V]
~I
nn-1
®F
I
-
nn- 1®F
to check that s is a morphism of groups.
s is an epimorphism.
_
~2
------
=
exact.
V ® w(b ) bounds in
However ?>[8M] = 0 in I·
nn- 1 ® F 0 because
it is the bordism class of the boundary of the complement of a regular
-
-
neighbourhood of 8M in M. The next lemma is important in what follows. Lemma 2.8. dary
Suppose that M is a
(p,
n+1)-manifold with boun-
V and label it by b
;
with singularities V, representing
an element in H (p,
W,
58
N has singularities
then
V,
I
cp 2'
up to codimension p + h + 1 at most, at most.
N:
of
with
W
to each manifold
nn- 1) we are able to associate a nn(p), such that
representing an element in In fact, if W'
with boundary, W, OW,between M, oM and N, oN such that
(c)
E BI;
nn-1).'
identify each copy with V ® bl•
(p,__ n)-manifold
exists a bordism
OW has singularities
I
in codimension one only. Therefore,
that W' has singularities
(b)
H (p,
the above identification on its boundary, becomes a (p, n)-manifold
has singularities up to codimension p + h, there is a (P, n)-bordism W has singularities
E
n n- 1 ® F 0, i. e. V ® w(bI) = aV. Take a copy aV consists of a number of copies of V (non
constantly labelled in general);
oM. Then, if oM has singularities up to codimension p and M s[W] ~ [V] + im
(a)
2
O. 8uppose V constantly labelled by b l
is not necessarily
Take [V] + im?>
l
l
0
n n- 1)
[8M] + im ~ .
It is straightforward
the sequence
Thus there is a well defined
map: s : nn (P)
Fig. 11
1m ~2'
E
E
[W] we can assume, by Lemma 2. 8,
8W' in codimension at most one and that there
W' - VI with singularities
8N in codimension two
Then [sW] - [8W'] = ¢2[8W] and so s is an epimorphism.
up to codimension p, up to codimension p + 1.
59
running:
Description of the map l Define a map I: defined homomorphism; I
l
fiJ
1
n
® F
..• n 1
= 0 because let
by bI E BI.
[M]; l
is a well
®
non
F
[W] =
..• n
mula when
(p).
lli> 1[M] and suppose M constantly labelled p)
and observe that
of W to the empty set by means of a
(p,
n+1)-manifold with codimension
n*(-;p).
Pick a representative
O.
N, of V to
l/J
is exact.
p
3. FUNCTORIALITY The classes of links constructed in Section 2 summarize the whole structure of the resolution we refer to
one singularities. there exists a bordism,
(-))=}
a(Mx bIL(bJ,P
is bordant to W. So stick the two bordisms together and get a bordism
=
q
This spectral sequence collapses to the universal coefficient for-
Then take M x bIL(bI,
Assume now l([M])
n
H (p, p
l
l/J
n
=
® F ..• n (p) by l[M] non so we have the sequence
n
If
V of l([M]):
p
P,
p'
map f : p .•• p'
such that N has singularities
p
geometrically.
Therefore,
from now on,
as a linked resolution. are linked resolutions of G, G' respectively, is said to be a map of linked resolutions
a chain
(or simply
linked map) if f is based and link preserving, i. e. :(a) f(bP) E B'P for each bP E BP. remove from N a regular neighbourhood of SN in N to get the required (b) Let bP E BP. If we relabel each stratum of the link bordism between M and Ii> (SN). Thus we have proved that the sequence 1 L(tP, p) according to f and if f(L(bP, p)) denotes the resulting object, above is exact; which is enough to ensure the existence of a monomorthen f(L(bP, p)) = L(fbP, p'). phism l : H (p, n ) ..• n (P) induced by l. ann So there is a category, e, whose objects are linked resolutions Now it only remains to prove exactness at n (P). ---n E P ••• G and whose morphisms are linked maps. If Cib* is the category sl = 0 : sl[M] = 0, because [M] has no singularities; hence We claim that ~ [SN] = [M]. In fact
SN in codimension one at most.
sl
=
1
of graded abelian groups, we have the following
O.
Ker s
C
im l:
let [M] E n (P) and assume, without loss of n
generality, that M has codimension one singularities means that [SM] E im dism; therefore
l/J. 2
C
im
p
[M] = [M'] where M' is without singularities
l = im t.
=
n
® F
a
whose image through
l
Proposition 3.1.
0
But then SM can be re-solved up to bor-
determines an element of n Thus Ker s
SM; s[M]
sake of simplicity wedisregard
and hence
is [M].
Proof.
[M] ..• [T(M)] p
We have seen how the exactness of p is used in
the proof of the universal-coefficient
theorem.
As pointed out before, if p is any based ordered chain complex augmented over G, then the theory way. But now the singularities necessarily
60
solvable;
the topological component of n*(-; -).)
Let T: p .•• p' be a morphism of
we associate a (p', n)-manifold,
the
e.
If [M] En (-; p), p
T(M), to M by relabelling
n
all the strata
of M according to the based map T. The correspondence
The proof of the proposition is now complete. Remark 2. 9.
e ..•CLb*. (For
n*(X, A; p) is a functor
p
T* : n*(-;
p) .••
gives a well defined natural transformation n*(-;
p')
Corollary 3. 2.
and the functorial properties If the linked map T: p .•• p'
of theories
are clear. is a homotopy equi-
valence, then T* is an isomorphism.
n*(-, p) can be defined in the same
in codimension greater than one are not 2
Proof.
they give rise to the E -term of a spectral sequenC11 theorem.
This is an easy consequence of the universal-coefficient
There is a commutative diagram
61
0-
H
° (p, ° (p',
j
AI)j
H,(T, 0n(X, 0'" H
Obviously the above definition of
0 (X, A))'" 0 (X, A; p) - H (p, 0 l(X, A)) - 0 n n 1 n-
0 (X, A)) n
T.
H, (T, 0n_1 (X,
lX, A;
~i
p) •••
H
j)
0
(p, 1
n-
A)I
I
p'
E
e.
nn (X, A; p)
is independent of the chosen
Now we fix our attention on a particular the canonical resolution of G and written
l(X, A)) - 0
y.
t. 1. resolution,
called
It is defined as follows:
£
in which the side-morphisms equivalence.
Therefore
T
are isomorphisms,
*
because
T
y:r
is a homoto
is also an isomorphism.
If G E
-r,
•
r-+-G~O
\~2/0 ~I/a: '\
(t. 1. ) resolution of G is an exact
2
FKer£
sequence F
2
-F
where 1
-
F
°
r , F Ker £, F Ker Ker lfi
satisfying conditions (a), (b), (c), (d) in the definitions of structured resolution given in Section 2. defined as in the non-truncated objects are t. 1. resolutions
A linked map between t. 1. resolutions case and there is a category,
e,
{n*(x, A; p)}
on CP x ()) (G = category of topological pairs). from p by choosing a based kernel of lfi.
2
(X, A) is a pair
1
is the obvious map;
r = FBI 1
iJ f in terms of the elements of G 1
(X, A),
r
Fix a p' E e obtained
2
~
2
M and a map
r };
=
a 2 (h),
expressing
A singular (P, n)-cycle in
C
°
f and lfil = 0I~I = FB2; B2 = {h; w(h), c(h) Ih E Ker lfi , w(h) is a word
~I(f, w(f))
which is a functor
(M, f) consisting of a (p', n)-manifold
1 1
are the free abelian groups on G, Ker E,
lfi
B = {(f, w(f)) If E Ker E C F Ker E and w(f) is a word expressing
and whose morphisms are linked maps.
we construct a graded abelian group
1
a
is
whose
In the following, for each p E ~ and each topological pair
°
respectively
1
c(h) is a cancellation rule associated to h}.
~
and
2
are defined similarly y has canonical bases
2
G, BI, B
and a canonical structure f : (M, oM) - (X, A) such that M has at most two intrinsic strata labelled w hO h (h, w(h), c(h)) has (w(h), c(h)) as its structure. 1C I by elements of B and B . A singular (p, n)-cycle (M, f) is said to
in
°
bord if there exists a (p', n+l)-manifold
Wand a map F : W-X
!!
Lemma 3.3.
for
lfi:
G ..• G' is a homomorphism of abelian groups,
is a t. 1. resolution of G, y' is the canonical resolution
which (a)
M is a submanifold of oW
(b)
FIM
=f
and F(OW\M)
C
lfi extends to a linked map ~: p -
A
(c) M has at most three intrinsic strata (labelled by elements of BO, BI, B2).
y'
of G';
in a canonical way.
= {F , I/>}, y' = {r', lfi' L We proceed by p p p P induction on p. Write (~) = (~o' ~I' ~/ For p = 0, put ~o(bo)=I/>E(bo) Proof.
°
Let
°
p
~
Define -(M, f) = (-M, f). Two singular (p, n)- for each b E B. Inductively, let E B.p ~hen -I/>p_1I/>p(bP) EKerl/>~_l Ofth dO " "t " and therefore it determines a basis element, b , in F Ker I/>p' 1; b'P has cyc 1es (M , f) , (M , f) are b or dan t 1 e 1Sl01l1umon 2 " (X A) B d" ° ° 1 1 t" a canonical word w(b'P) and cancellation rule c(b'P) induced from those (M1 U - M2'1 f 1 IUf 2 ) ~ or ds 111 , . or Ism IS an equ1va ence re a lOn .p __ of 0- through the map (I/> lfi) Therefore the assignment in the set of singular (p, n)-cycles of (X, A). Denote the bordism class of p-l' p-£' tfl- (b'P " w(b'P) c(b'P)) defines lfip with the required properties. (M, f) by [M, f] and the set of all such bordism classes by On(X, A; p), ,
W is called a bordism.
An abelian-group structure
62
is given in Un(X, A; p) by disjoint union.
I
63
Lemma 3.4. Proof. If
n*(x, A;
Functoriality
gives a functor
p)
on
<.P
is a morphism of
T:p ....•p'
x
<.P
e .• Cib*,
Theorem 3.6.
is obvious,
e
and [M]
p
d~(-;p), n
to T. The correspondence
-
-
[M] .• [T(M)1
P
-
To each G
E
Gb assign
n* (X, A;
morphism
G
Let
be an i-canonical resolution for i
Pi
=
2, 3. Then
gives the required natural trans- there is a commutative diagram
:~~~F:::or~i:' 0:; ;1 ::::i~~ndorWx resolution of G; to each morphism
Proof.
S0.0<,~~, -7,,~;
P
p) ....• n*(-; p').
G
let T(M) be
the (p', n)-manifold associated to M by relrcbelling each stratum according formation T* : n*(-;
: n*(X, A; P ) ....•n*(X, A; G) is a natural
t
equivalence of functors.
y)
where
cJ>: G ....•G', cJ>
~* : n*(X, A; y) ....•n*(X, A; y'), where
y
tG
O.<X,
"b - db.,
t1
,
2
p,)
is the canonical
Q/ ~
n*(X, A; p )
Cib, assign the homo-
E
¢
is the canonical
3
where t ..
1, J
and Q/ are the natural transformations
obtained in the usual
extension of cJ> described in Lemma 3. 3 and ¢* is the induced homo-
way by relabelling the cycles according to the canonicallifting's
morphism described in Lemma 3.4.
id : G .• G. By Corollary 3.2, t. . is an isomorphism 1,
In view of the previous corollary we shall write n* (X, A; G)
Therefore,
instead of O(X, A; y) and cJ>* instead of ¢*. A (y, n)-cycle [bordism]
in order to prove the theorem,
<
j :5 3).
we only need to show that
Q/
is an isomorphism.
will also be called a (G, n)-manifold [(G, n)-bordism].
Q/ is an epimorphism:
A linked resolution of an abelian group G
J
of
(1:5 i
fact that t
1, 3
it follows from commutativity and the
is epi.
Q/ is a monomorphism: let Mn be a (singular) G-manifold such that Q/(Mn)_ ¢ in n (X, A; p). Then Mn determines an element n 3 [M] E n (X, A; P ) such that t [M] = O. Since t is a monois said to be p-canonical (a)
F
p-
1" ....•
length p - 1, i. e.
E
G .• 0 is the canonical resolution of G of
= r., 1 1
F.
(b) Fp+1 = Fp+2 Let G E Cib and P
n
2
O:s i :5 P - 1, and the morphisms
=
G
E
cJ>. are 1
y.
O.
e
we observe that p
P2
2,3
morphism, we deduce that there is a
the same as in the definition of
F 2 = F 3 = 0).
P2
(1:S p :5 3) if
p
'+ y
2
2,3p
bordism
n
W: M
Then we have the homology theory
{n*(X, A; PG)'
the possibility of
making bordism with coefficients in a short linked resolution
a]
ro. Finally
is a linked embedding of resolutions and therefore W provides the required bordism of Mn to zero in 0 (X, A; G). n The proof of the theorem is now complete. 2
We are now able to state the theorem asserting a short linked resolution of G, (i. e.
~
E
p ....•G
depend functorially on G.
defined in Section 1. We also have the graded functor 0* (X,A;G) : <.P •••.•
Cib* constructed at the beginning of this section.
It follows from
Theorem 3. 7.
Lemma 3. 3 that there is a canonical extension of the id: G •.•G to a linked map id: P
G
....•y.
The latter induces a natural transformation
functors t : n*(X, A; P ) : O*(X, A; G) obtained by relabelling the G G PG-manifolds according to id. The next theorem is the main step towards functoriality.
U*(X, A; G) : of
(a)
There exists a (graded) functor
x Cib'" Cib* which associates to each pair
the graded abelian group n*(X, A;
PG)
l,presentation of G; to every morphism .
the graded homomorphIsm (b) In ,aturally
64
<.P
where (f,
cJ»
P
G
:
(X, A; G)
is a fixed linked
(X, A; G) ....•(X', A', G')
-1
(f*, t , cJ>*t ) : n*(X, A; P ) .• n*(X, A; P ,). G G G G Functors corresponding to different choices of P are G
equivalent. 65
The result follows immediately front Theorem 3.6 and Corollary
theory n*(-;
With the notations of the theorem, we define n*(-; G), the p.l.
particular,
3.5.
pl. Some of the facts about morphisms, that we have
established in this section, hold in the case of chain complexes.
oriented _bordism with coefficient group G, by n*(X; G) = n*(X;
P ). G
if T: p -+ p'
complex and p'
is a linked resolution,
then the proof of Proposition
applies to give an associated morphism Corollary 3. 8.
For every pair
X, A, every n
2':
0 and every
abelian group G, there is a short exact seguence
o -+ G @
which is natural in (X, A) and in G.
set up, namely n*(X, A; features:
versal coefficient sequence associated to n*(-; G). Precisely,
since
ab, Hilton [2; Theorem 3.2],
the former is readily seen to be a generalized homology theory; Theorem 3. 6
between the two functors, which G ~roves at the same time that n*(X, A; P ) is natural on ab and that
t
G
1l*(X,A; G) is a homology theory. 1l*(X, A;
For every abelian group G, the universal-
n*(X, A;
P ), G
context (i
=
= 1
1, 2; P.
sequence of nn (-; G) is pure.
Corollary 3.10.
For every pair
S1(X,A; G) is more appropriate
Pi)'
to the
any i-canonical resolution of G).
4. PRODUCTS
From algebra we deduce:
2':
In the following we may use whichever of the equivalent functors
Corollary 3. 9.
integer n
as
and O*(X, A; G). They have different
PG)
establishes a natural equivalence
gives us the following
co~fic~nt
3. 1
p) -+ n*(-; p').
while the latter is natural on the category of abelian groups.
We are now able to say something about the splitting of the unithe sequence is natural on the category
T*: rl*(-;
The above treatment of functoriality can be summarized follows. For every abelian group G, two functors
n (X, A) -+ n (X, A, G) -+ Tor(G, n l(X, A)) -+ 0 n n n-
In
is a linked chain map, P is a linked chain
(X, A), abelian group G and
0 such that Tor(nn_1 (X, A), G) is adirect
If G, G' are abelian groups, let P be a linked resolution
sum of cyclic O-+F
groups, the universal coefficient seq~ence
E
-+ F
-+ G-+O
1
0
with 0-+ n (X, A) ® G -+ n (X, A, G) -+ Tor(rl n
n
~
l(X, A), G) -+ 0
O
B
splits.
1
B
The class of examples of splitting considered by the previous corollary is quite vast, because it includes the following cases:
IF",
(a)
any G finitely generated
(b)
any G such that its torsion subgroup has finite exponent
(c)
any n
n-
and p'
G
=
{gl'
g2'
{r l' r 2'
••.
defined similarly.
••• }
Then p
is the augmented chain complex
2
0-+ F
@ F' 11011000
.•• F
1
@
F' tB F
@
F'
.•• F
E" @
F'
-+ G
@
G' .•• 0
where 1J!"(r@ r')
=
(p, n)-manifold makes sense in the case of P being any linked chain
=
g
complex (not necessarily
=
As we have pointed out earlier,
the definition of
a resolution) and there is an associated bordism
2
1
1
E"
66
p'
1J!"(g@ r')
Remark 3.11.
@
1(X, A) such that its torsion subgroup has finite
exponent.
= =
=
@
@
E @ E'
67
p @ p'
is based by means of B\ B,i (i = 0, 1); it is structured
dimension one by the structures assign to r @r'
E
of p and p';
B,2 the word w(r
®
in
in dimension two we
r') = w(r) @r' - r @w(r').
For
now we do not fix a cancellation rule. Let M be a
(p,
i)-manifold with singularities
j)-manifold with singularities
(p',
M x M'.
SM'.
SM; M' a
Form the cartesian
product SM
It has three intrinsic strata given by
SM'
r
g' 2
(M - 8M) x (M' - SM') SM x (M' - SM')
U
(M - SM) x SM'
SM x SM' Fig. 12
On each stratum we can put labels via the tensor product, i. e. if V is a component labelled by x : V' labelled by x', V x V' is labelled by x
@
x'. We show that M x M', with such additional structure,
I
is a
I..•. I
i + j)-manifold. The first and the second stratum are easily seen to be p @p'-manifolds of the appropriate dimensions and we are
(p ® p',
going to examine the third stratum.
For simplicity assume
1.1'.',
~,
SM, SM' , I
constantly labelled by r, r'
respectively,
so that SM x SM' is labell1
g7;t" g;
by r ® r' E: B,,2. The basic link of SM x SM' in M x M' is topologiq the join L" = L(r, p)*L(r', is a
(p
@p',
L" with the structure
p').
I)-manifold because
M x M' - SM x SM' is a
manifold and there is a product structure where @ is meant to act on the labels. Thus L" is a (p @ p',
dimensional stratum represents is a union of disks. assigned to r @ r'
Therefore
The zerof
@r' Ur@-L(r',
But this represents
\;
w(r @r')
d
2
3
r® g'
3
g
i
I8l 1
r'
g ® g'
pi
1
t2~.~
stratI I
L(r @r', p @p'). See Flg,.
1
,
and the I-dimensional
i+j)-manilold.
g'
I8l
1@V:,\~
L" gives a unique cancellation rule to ~
=
-r
the word :
I)-manifold in which the zero-
in order to' have L"
M x M' ;., then a (p "p',
p)
11'
(p ® p')-
around SM x SM'.
dimensional stratum of L" is isomorphic to L(r, w(r ® r').
induced by M
:'" g~'
Picture of a neighbourhood of SM x (M' - SM') in M x M'
, 12 and 13.\
J
r
I8l
r
I8l
g'
2
basic link of a point of SM x SM' in M x M'
'.}
g
@
3
r'
g
I8l
3
g'
3
3
Fig. 13 68
g'
69
We can now define a homomorphism x
p, p'
: n (-' *'
p) ®
n (-' *'
5. THE BOCKSTEIN SEQUENCE
p') -
n (-' * '
p ® p')
Theorem 5. 1.
On the category of short exact sequences of
abelian groups
by
o-
there is a natural connecting homomorphism
x p, p' is of degree zero. Let
p3
'" ljJ G' - G - G" - 0
be a 3-canonical resolution of _ G ® G'.
-
Then, by the
proof of 3. 3, there exists a canonical lifting id: p ® p' - P
-
(3 : n*(-; G") - n*(-; G')
of
3
id: G ® G' -G ® G'. Therefore we can define a cross--,pc:r:...;o:..:d::..:u::..:cc::. homomorphism't f dId o egree -
an
t 11 a na ura ong exact sequence
G , G' : n*(-; G) ® n*(-; G') - n*(-; G ® G')
X
by the composition Proof.
x
n*(-; p) ® n*(-; p')
G1.P~ __
~
n*(x
where PG®G'
p,p'
.'
is a linked presentation
;lP~~G') t1,
under the assumptions:
of G ® G' and
C
G and (X, A) = (point, ¢).
sarily exact) sequence of canonical resolutions
r'2 '+ r 2 ~
is the usual
r'1
relabelling map (as in the proof of 3. 6).
e-.
~
r1 ~
~
Remark 4. 1.
G'
Realize the exact sequence of abelian groups by the (not neces-
3
n*(-; P3)
id*
For the sake of clarity of exposition we prove the theorem
r'0 '+ r 0
If an abelian group G is also endowed with a multi
plication that makes it into a ring, then we have a product homomorphism:
l G'
o -
~
'+
G
-* - -
and linked maps
r"2 ~
r"1
r"0
l
G"
0
(I) Definition of (3 n
Let M
be a G"-manifold.
Suppose that the singularities
ave only one connected component V ® r", with r" relation in G". where x G G is the cross-product
,
m(g
Q\)
g')
= gg'.
and m: G ® G - G is given by:
The homomorphism
ring and if (X, A) is a pair,
Il makes
1
n*(point; G) into a
n*(X, A; G) can be given a structure
graded module over the ring n*(point; G) in the usual way.
70
, ••• ,
of
(gl + ...
,=
gt
We relabel
=
g~ + ...
V by an element of G' as follows.
of M
+ gt a Choose
G such that ljJ(g.) = g? and g. = g. ~ g? = g~'. Then 1 1 1 J 1 J + gt) = g; + '" + gi' = 0 in G". Therefore by exactness E
L.g. is an element of G'. 1 1
We relabel
V by g' and get a
G', n-l)- manifold V Q\) g'.
71
Now suppose g], ... , gt is another lifting of g~, ... , gi' as above, giving a (G', n-l)-manifold
V®
g'. We show that V
V ®g' are bordant as (G', n-l)-manifolds. g~' into g~ = g. - g. 1
1
1
EO
1
G', i
=
(g' - g') is a relation in G'.
® g'
and
Relabel M \V by changing
1, ... , 1. The sum r' = (g' + ... 1
a 'relation amongst relations'
the common part
If (3(W)
!3(M)- 0.
Now we are entitled to define
+ gt')'
Therefore take a labelled copy V ® r'
V x cone(g~ + ...
Tn-1 by r'.
denotes SW with the relabelling described above, then !3(W) provides the required G'-bordism
!3 :
of V and form the polyhedron W = (V ® r') x L(r', G') UM, where L(r', G') is the link generated by r'
of G". We label
-n (-' G") -
-n
n '
n-l
(-''
G')
in G' and the union is taken along·by
+ gP (see Fig. 14). W is a
(G', n)-manifold and provides the required bordism between V ® g' and V
®g'.
(2) Exactness at 0 (-; G) n
(a)
lJI*lJl* =
is a (0, n)-manifold. g' V®r
I
1
g
Ker lJI*
C
then M"=lJIf/>(M')
M" - 0 by the proof of the universaln
1m lJl*. Let M
(G", n+l)-manifold with
be a G-manifold and W" a
aw"
= lJI(M). We show how to modify W" in order to get a G-bordism between Mn and a G'-manifold M,n.
3
Ck;J g
If M' is a (G', n)-manifold,
Therefore
coefficient theorem.
g'
(b) g
o.
Relabel each component of the (n+I)-stratum
g' 2
3
of G, obtained from the G"-labels through a lifting
2
(a)
The relabelled n-stratum stratum of Mn.
Fig. 14
(b) If the singular set, S(M), of M has more than one component, the relabelling construction can be performed componentwise and one
of
aw"
of W" by elements G t G" such that
..•.-
-
coincides with the n-
If two components are labelled by the same element of G",
the corresponding liftings coincide. Let V be a component of the n-stratum V
of W" and g , ... , g 1
V
gets a (G', n-l)-manifold, !3(M), whose bordism class is independent of the new G-labels around V; g' = ~ g. is an element of G'. Attach a the various choices. i=l 1 Next we show that !3(M) depends only on the bordism class of new sheet (V x I) ® g' to V iff g' '" 0 and label V by the G-relation M, i. e. if M
<2" (0,
aw
= M.
g
G such that lJI(g)= g" and relabel
EO
r, ..,
Let W be a (G", n+1)-manifold wi r = Ligi - g' (see Fig. 15). Now let V ® r, V ® r, V ® be the is a component labelled by g" EO G", choose components of the relabelled n-stratum merging into a component T of
then !3(M)~' 0.
If V C W\f3W
V by g. Let Tn be a com-
the (n-l)-stratum. The corresponding new sheets which have been inserted, namely (V x I) ® g', (V x I) ® g', (V ® I) x =g', •.. , are, by
ponent of the n-stratum of W. The sum in G of the new labels on the sheets coming into Tn is an element g' of G'. We relabel T by g'. construction, Finally, if Tn-l
is a component of the (n-l)-stratum
Tn ® g', ... , Tn ® g' lIS
then
S
r' = ~ g: is a relation in G', because i=ll 72
of Wand
are the sheets merging into Tn-I, Tn-l
such that r' = g' + g' +
g' +
...
is a relation in G'.
Therefore we can glue them to one another along a new n-dimensional sheet (T x I) ® r'. G-bordism.
The resulting polyhedron
W provides the required
was originally labelled by
73
(ii)
=
r
In M" we replace each g~' by g. and V ® r" I
I
by V ® r, where
g' - L.g .. I I
(iii)
We attach W' to the relabelled
M" identifying
aw' with V®r.
It is readily checked that the resulting labelled polyhedron is a (closed) G-manifold such that lJI(M) G" - M". See Fig. 16
Mn
g"
C>C) g~
g~
Fig. 16 (4) Exactness at 0n(-; G')
q,*f3
(a) singularities
= O. Let M"
n+1
be a G"-manifold with connected
V ® r", r" = L.g~'. Then f3(M")= V ® g' where I I
g' = L .g. and lJIg. = g~. We construct a G-bordism
Fig. 15
I I
I
I
W: (3(M")- yf as
follows. (i)
(3) Exactness at 0n(-; G")
(a)
n
{3lJ1* = O. Let M
r
be a G-manifold.
definition of (3, we have (3lJ1(M) ='---'V
According to the
® 0, where V varies over the
V
We relabel
= L I.g.I -
(ii)
M by changing each g~ into g. and r" into I
I
g'.
We attach a new sheet (V x I) ® g' to the relabelled
the singularities
V ® r.
M" along
See Fig. 17
set of components of S(M). But V ® 0 - ¢ by a trivial G' - bordism. (b)
Ker {3C 1m lJI*.
Let M"n be a G"-manifold.
sake of simplicity let us assume that the singularities
For the
of M" have only
one component V ® r"; r" = L. g~. Then (3(M")= V ® g', where I
I
g'::;: L.g., lJIg. = g~. By assumption there is a G'-bordism I I I I We construct a G-manifold Mn as follows. (i)
Since {3(M") has no singularities
singularities
W' : (3(M")-~,
we can assume that W' has
in codimension one at most, because otherwise we solve
Fig. 17
the codimension-two stratum as in the proof of the universal-coefficient theorem. 74
75
(b) a G-bordism,
Ker
q,* c 1m p, Let M,n be a G'-manifold and W: M,n -
From
W we get a G"-manifold W" of dimension
¢
(n + 1) 6.
as follows, (i)
The proof of exactness is now complete and naturality is clear. BORDISM WITH COEFFICIENTS
We remove from W all the strata which are labelled by elements
of G' or by relations or by 'relations amongst relations', resulting object W" is a (G", n+ I)-manifold with singularities
aw =
dimension 52 and it is closed because
In order to define coefficients in an R-module we need the following additivity lemma.
We relabel the remaining strata according to the map 1/1.
(ii)
The
in co-
M' has been removed in
Lemma 6. 1. then
We get W" by re-solving the codimension-two singularities
W" up to a bordism which has singularities in codimension Now we show that (3(W") is G'-bordant to M', singular part of the bordism used in (iii).
+ (3)*=
«(II
step (i), (iii)
--
Proof.
of
(11*
!!
(II,
+ f3* :
Consider the chain map
Let Q" be the
=
put I/I(M)
(II(M) + f3(M) - «(II + (3)(M).
is labelled by 'relations amongst relations';
D: 1/1"" O.
(3(Q") (constructed as in
The resulting object W
(i)
is not a o into the required
Let V ® r be a component of the n-dimensional stratum of W,
with r = g' + . " + g' + g +, .. + g (g~ E G'; g. E G - G'). Then 1 p 1 q 1 J V ® g' C (3(W") where g' = g +." + g, Therefore we attach a sheet q
1
(V x I) ® g' to W r'
=
g' +", ]
along V and change the label r into a + g' + g', which is now a relation in G', p
--
Let V ® r, V x
(ii)
component 'I'
®;
r, V
®
r, '"
...
q,l
r
2
/
1
r'
...
r'
l/~;h{"
G-bordism Wand remove from it all the top dimensional strata which We show how to make W a G'-bordism by inserting new sheets.
...
q,2
r
N'
To this purpose we reconsider the
G'-manifold in general.
-
If [M] dl n(-; G),
Then I/I(M) is a (G', n)-manifold
Un(-; G'). But ~ is a lifting Therefore there exists a chain homotopy
-
the proof that the Bockstein is well defined) realizes a G'-bordism between
are not labelled by elements of G',
~
«(II + (3) where
and we only need to prove that ~(M) - ¢ in
Q" has at most two sheets;
the non-singular one is labelled by relations in G" and the singular one
M',
- = -(II + {3-
1/1
(II, {3, are the canonical - (II + {3__ ~ liftings of (II, {3, _ (II + f3.
53.
(3(W") and {3(W"), Therefore we only need to provide a G'-bordism
{3: G'" G' are abelian-group homomorphisms,
U*(-; G)'" U*(-; G'),
of the zero map 0: G'" G',
between (3(W") and the original
IN AN R-MODULE
2
so that
1/1 1/1
1
=
1
...
e;
r
.•• 0
a
/
...
G
1~"... 1° r'
G'
a
.•• 0
Da q, 1 + f/I'D 2 1
o =C/>'D. 1
0
By definition the singularities the induced diagram
of
1/1
(M) are given by
1/1 (SM).
Consider
be components merging into a
of the (n-l)-stratum,
where;
=
r +
r+r
+ '"
is
a relation amongst relations in G. The corresponding new sheets which have been inserted, are, by construction, G',
namely (V x I) ® g', (V x 1) ® such that r'
=
g' +
g', (V x I) ® g', ,."
g' + g' + , "
is a relation in
Therefore we can glue them together along the n-dimensional sheet
('I' x I) ® r',
The resulting polyhedron provides the required G'-bordism
N' : (3(W") - M',
76
77
-
where 1 ® D is now a homotopy of 1 ® ljI to zero. represents an element in n
-
-
ljI(SM)=(l®ljI
But _
1
n-
=
(1
-
of
ljI(SM)
-
As we know, SM
0
(1
®
® CP). 1
a" is the R-module structure of G. The pair
)0 (l®CP)(SM)+(l®lJl')0 (l®D
=
where
So we have
[SM] = Ker(l
lJl')(W'), where W'
®
universal-coefficient larities
1
)(SM)=(l®D
(1 ®lJl)(SM)- 9f because
ljI(SM)
1(-) ® r.
module G'. 1
)(SM).
Therefore
{n*(-; G), a}
The structure
is 'bordism with coefficient~}n the R-
a will be dropped from the notations.
If f : G'" G' is an R-homomorphism,
then for every r
E
R we
have commutative diagrams f*
D )(SM). By the proof of the
n*(-; G) ---.-
1
n*(-; G')
theorem this is sufficient to ensure that the singua(r)!
can be resolved by a bordism of (G', n)-
ljI(M)
manifolds.
So, using the homotopy D, we have eliminated the singular
stratum of
ljI(M).
G
•• G'
n*(-; G)
----1._
a(r)
n*(-; G')
Let V' be the resulting (G', n)-manifold.
a_ component labelled by g'0
E
G', then there exists
g0
E
If V'o is G such that
func~:riality!
Hence n*(-; G) is a functor on the category of R-modules and R-homomorphism.
From the naturality of the Bockstein sequence for abelian
(g ) = g', because the process of resolving the singularities in ljI(M) o 0 0 does not change the labelling of the top dimensional stratum. Therefore
groups, it follows that there is a functorial Bockstein sequence in the
the element of n (-) ® r' represented by V' is the image of some
category of R-modules.
ljI
[V]
E
n (-) ® n V'
=
r
n
0
ljI(V)
0_
through 1 ® ljI.
=
Then again we have
0
(1 ® '/>')(1 ® D )(V)
o
Summing up, we have the following:
(a)
n*(-; G) is a functor on the category of R-modules
(b)
n*(-; G) is additive
(c)
For every short exact sequence of R-modules,
there is an
associated functorial Bockstein sequence. so that V' may be borded to 9f by a (G', n+1)-manifold with singularities given by (1 ® D )(V). o We now turn to the main object of this section, i. e. putting coefficients in an R-module.
Properties
(a), (b), (c) form the hypothesis of Dold's Universal-
coefficient theorem [1]. Therefore we deduce that there is a spectral sequence running
In the following R will be a commutative 2
E p,q
ring with unit.
=
Tor (n (-'R) G) =} p q' , p
n (-'G) *'
If G is an R-module, let n*(-; G) be bordism with coefficients This completes the discussion of the case of R- modules as co-
in the underlying abelian group G; n*(-; G) has a natural R-module structure.
In fact, we must exhibit a ring homomorphism
a: R ..• HomZ(n*(-; G), n*(-; G)). The above additivity lemma, together with functoriality,
tells us that there is a ring homomorphism
efficients.
In later chapters we shall only deal with abelian groups;
but
it is understood that everything we say continues to work in the category of R- modules. REFERENCES FOR CHAPTER ill
defined by a'(f) = f*.
Therefore we can define a by the composition
[1]
Math. Zeitschrift,
80
(1962/3).
a
R ..• HO~(n*(-;
G), n*(-; G))
a~ la' Horn (G, G) Z 78
A. Dold. Universelle Koeffizienten.
[2]
P. J. Hilton. Putting coefficients into a cohomology theory. Konikl. Nederl. Akademie van Weterschappen (Amsterdam), Proceedings,
Series A, 73 No. 3 and Indag. Math. 30 No.3,
(1970), 196-216. 79
P. J. Hilton and A. Deleanu.
[3]
efficient sequences. (1970),
[4]
On the splitting of universal co-
Aarhus Univ., Algebraic topology Vol. I,
180-201.
J. Morgan and P. Sullivan.
The transversality
class and linking cycles in surgery theory. (1974),
IV· Geometric theories
characteristic
Ann. of Math. 99
463-544.
In this chapter we extend the notion of a geometric homology and cohomology (mock bundle) theory by allowing (1)
singularities
(2)
labellings
(3)
restrictions
on normal bundles.
The final notion of a 'geometric theory' is in fact sufficiently general to include all theories (this being the main result of Chapter Vn). A further extension, to equivariant theories,
will be covered in Chapter V.
In the present chapter, we also deal with coefficients in an arbitrary geometric theory.
A geometric theory with coefficients is itself
an example of a geometric theory and it is thus possible to introduce coefficients repeatedly! The chapter is organised as follows.
In
§1
we extend the treat-
ment of coefficients in the last chapter to cover oriented mock bundles and in
§§
2 and 3 we deal with singularities
and restrictions
on the normal
bundle. In §§4 and 5 we give interesting examples of geometric theories, including Sullivan's description of K-theory [11] and some theories which represent (ordinary) Z -homology. p
Finally
§6
deals with coefficients
in the general theory. 1. COBORDISMWITH COEFFICIENTS We now combine Chapters n and III to give a geometric description of cobordism with coefficients.
It is first necessary to introduce oriented
mock bundles (the theory dual to oriented bordism).
We give here the
simplest definition of orientation, an alternative definition will be given in §2.
n n Suppose M , V -1 are oriented manifolds with V
C
aM. Then
we define the incidence number £(V, M) = ±1 by comparing the orienta-
80
81
2.
tion of V with that induced on V from M (the induced orientation of aM is defined by taking the inward normal last);
orientations agree and -1 if they disagree.
E(V, M) = + 1 if these
An priented cell complex K
<
an
E
K.
I K'
An oriented mock bundle is a mock bundle UK in which each we have E(~(T),
~(a))
=
E(T, a).
plicial, then f-l(A) that, if f: E
block is oriented, K is oriented and such that, for each Tn-l
<
an
E K,
is collared in f-l(A*)
i. e. if f: J for each A
E
-+
K is sim-
K. Here A*
-+
K is a simplicial map, E is a (p, n+q)-manifold and
is an n-manifold, then the inverse image of a dual cell in K cuts
the singularities
of E transversally,
so that f: E
-+
K can be made
into the projection of a (p, q)-mock bundle.
We leave the reader to check that the
theory of oriented mock bundles enjoys all the properties
theorem in its full generality,
is the dual cone of A in K with base A. From this theorem it follows
is a cell complex in which each cell is oriented and then we have the incidence number E( T, a) defined for Tn-l
In order to prove Poincare duality, one needs Cohen's
transversality
of the unoriented
3.
A discussion completely analogous to that of ill §4 can be
theory in Chapter II (the Thom isomorphism theorem holds for oriented
carried out. In particular there are functors
bundles and Poincare duality for oriented manifolds) - more general argu-
category of abelian groups and there is a universal coefficient sequence
ments will in fact be given in §2. This theory will be denoted n*(,)
place of ordinary manifolds. UK is a polyhedron
each
E
(Ji
K, p-l(a)
More precisely,
E(~) with projection
by using p-manifolds in
p)
a (p, q)-mock bundle
p: E(~) -+ K such that, for
is a (p, q+i)-manifold with boundary p-l(&), called
the block over a and denoted ~(a), and such that E(~(T),
~(a))
=
E(T, a)
n for each T < a E K. Note, E(V, M) is defined for p-manifolds V - \ Mn only when either V or -V c aM as p-manifolds (i. e. the inclusion respects the labellings, E(V, M)
=
orientations and extra structure),
+1 in the first case and -1 in the second.
-V
then denotes the
p-manifold obtained from V by reversing aU the orientations. It follows from the arguments in Chapters I and II that n*(, ; p) is a COhomologytheory, dual to the theory
n*(,;
p)
G) natural on the
and
the dual bordism theory n*(,). Now let G be an Abelian group and p a structured resolution of G. We define the mock bundle theory n*(,;
n*(,;
defined in Chapter
1lI, and from the arguments in III §4 that n*(p, Q; -) determines a functor on the category of abelian groups. We will leave most of the details to the reader and make some
also natural on the category of abelian groups. 4. If ¢: G ® G' -+ G" is a pairing, the cup product q r n ( ; G) ® n ( ; G') -+ nq+r ( ; G") and the cap product nq( ; G) ® nr( ; G') •.•nq+r( ; G") are defined using the usual pull-back construction and the cross product defined in Chapter m. 2. RESTRICTIONS
In this section we consider geometric (co)-homology theories which can be obtained from pi
of this is identical to the proof in Chapter il - the required extra struc-
(co)-bordism by restricting
bundles of the manifolds considered.
the normal
We sketch the case of cobordism.
Details for the bordism case may be found in p3; Chapter ill. p
Let E(~) -+ K be a mock bundle projection, an embedding i: E(~) -+ K x R normal block bundle V~IE(~) bundle map
remarks about some of the more delicate situations: Remarks 1. 1. 1. If ~;K is a (p, q)- mock bundle and I K I is an (oriented) n-manifold, then E(~) is a (p, n+q)-manifold. The proof
ON NORMAL BUNDLES
00
so that p
=
11
0
1
on E(~) in K x ROO.
then we can choose i, we then have a stable There is a classifying
~ TJ~
~
E(v ~)
E(y) U
I)
E(~) TJ~
•• BPt
ture all comes automatically!
82
83
, j
I
where y/BPL is the classifying bundle for stable block bundles. Now suppose that we have a space X and a fibration
4.
I
f: X'" BPi. ,products.
Then an (X, f)-mock bundle is a mock bundle ~ togeAtherwith a stable normal block bundle v f a classifying bundle map (fJ f 71~) and a lift of fJ ~ in X:
5.
Stable cohomotopy. X is contractible. Poincare
Again we have
duality holds for lI-manifolds. Let S be a discrete set and let X = BPL x S
Labelling.
,and f the projecti~
Then a connected (X, f)-manifold is just a mani-
IIOldlabelled by an element of S. Any function S x S ..• S gives this theory products.
See also the remarks at the end of the next section.
3. SINGULARITIES
,.. BPL
E(~)
Our treatment of singularities
is similar to that worked out by
,Cookeand Sullivan (unpublished) or to be found in Stone [9]. The theory of (X, f)-mock bundles is set up in exactly the same way as the theory of bundles.
In order to have products one needs in
addition a commutative diagram
Suppose we are given a class isomorphism).
I -
BPL x
n
of (n-l)-polyhedra(closedunderpl
Then a closed £ -manifold is a polyhedron M each of n
whose links lies in £.
A theory of manifolds-with-singularity
n
consists
oC for each n = 0, 1, .. , which satisfies: n 1. each member of oC is a closed oC I-manifold n n2. SoC 1 c oC (1.e. the suspension of an (n-l)-link is an n-link)
ofa class
m
XxX
£
(2. 1)
f xf
n-
BPL
3.
n
CoC 1 n £ n-
n
=
0 (1.e. the cone on an (n-l)-link is never an
-link). where
E9
is the map given by Whitney sum.
Using diagram (2. 1) external
Then an oC-manifoldwith boundary is a polyhedron whose links lie
products can be defined by q~x71 = m
0 (q~ x q1))' Similarly cap products ither in oC or CoCn_l' Then the boundary consists of points whose n are defined with the corresponding bordism theory (maps of (X, f)-mani- inks lie in the latter class, and is itself a closed oC I-manifold. More-
n-
folds into the space) and the proof of Poincare duality (for (X, f)-mani-
ver the boundary is locally collared (since its links are cones) and
folds) needs little change.
ence collared [8; 2.25].
The proof of the Thorn isomorphism theorem
for bundles with stable lifts in X can also be readily modified.
Notice that axiom 3 is necessary to ensure that the boundary is
ell-defined. Axiom 2 ensures that if M is an oC I-manifold then 1 O' e ted theory X = BSPL and f is the nExamp 1es 2.,2 . n n .• x I is an .£n-manifold with boundary. Axiom 1 implies that a regular ~ natural map. This theory has products. See also the alternative deseighbourhood of a polyhedron in an £n-manifold is itself an oCn-manifold cription given in §l. ..ith boundary. ..
f
2.
Smooth cobordism.
f : X'" BPL is defined using
PD
X has the homotopy type of BO and as in [7; §O]. This again is a theory
At this point we can remark that a manifold with singularities III the geometry of an ordinary manifold which was used in setting up
with products .. 3.
ordism and cobordism (mock bundles) and we get homology and co-
Pl spin cobordism.
f is the covering map.
has
X is the double cover
0f
BSPL and
.f
Again we have products •.
omology theories
Tt,(,),
"-'
T£*( ,).
Moreover the proofs of the Poincare
uality and Thorn isomorphism theorems are unaltered.
Note however,
.ee below, that products are not defined in general (but cap product with
I 84
I
I
85
the fundami:mtalclass of ;, manifold (amalgamation) is
:ways defined).
the point. ,
Products
(§4 contains details of killing. )
To combine £-theory with the restriction
on the normal bundle
:,'Of the last section, it is necessary to use the notion of 'normal block Suppose M and N are closed £-maJiitolds then M x N is in general not an £-manifold. £
4.
n
*£ c£ q
However it is one if we have:
jthe usual properties - the class of bundles and manifolds for which a
(i. e. the join of two links is again a link). !theory has Thorn and Poincare isomorphisms depends on the stable
n+q
Then, with axiom 4, we have cup and cap products. generally if £, 'JIT and ~
are three theories and £
jrestrictions imposed on the normal bundles.
More
* 'JIT c ~ then we
have cup and cap products from £ and :JlL theory to ~ theory.
For
example, if S is ordinary bordism theory (i. e. Sn = {(n-l)-sphere})
*
then £
lbundle system' as in Stone [9]. The resulting theories then enjoy all
Rather than attempt a
.j1rormal analysis of this general setting, we will give several examples ,insubsequent sections, which should make the general properties lthese theories clear.
coefficients (all the structure of a manifold with coefficients
S c £ by axiom 2, so that, as remarked above, cup and cap
eluded in 'restriction
products with bordism or cobordism classes are always defined.
of
We already have the examples, in Chapter Iil, of p is in-
on the normal block bundle system') and in
Chapter VII, we will give a family of examples, generated by the killing Basic links
process of §4, below, which include all homology theories.
A subset ill of £ =
U
£n is basic if no link in ill is a suspension
Finally we remark that we have now arrived at the general notion
and each link in £ is isomorphic to a suspension of a link in ill.
ofa geometric theory, since 'labelling' is included in 'restriction
on
Jnormalbundle', see Example 2.2(5). Examples 3. 1.
1. £
n
is the class of (n-l)-spheres.
£0
=
sq-2
{p'),
*
The set of basic links
{P' J. 2.
Basic links are I £1
= {xix ~ SO or
(n points)
J, q
2:
l j
tioned above, this is ordinary bordism theory. here is
As men-
E
£
and (n points)
o (n points)
E
£. 1
J, £ = {xix ~ q
Thus q-l S or
11, KILLING AND K-THEORY
I
In this section we give the general description of 'killing' an
I
lelementof a theory and apply it to give a geometric description of
1
fonnected K-theory at odd primes due essentially to Sullivan [ll].
See
1.
.. ~lso Baas [15]. Killing is defined in the following generality: This theory is 'twisted Z -manifolds'. A mamfold m the theory 1 n n-l .! 1. U and V are geometric theories. is either locally an ordinary manifold or like R x C (n points). TillS'I' ( ) 2. M is a closed V, n -manifold. theory can be made into 'coefficients Z ' by adding orientations and an i n j 3. There is a natural way of regarding W x M as a V-manifold, untwisted neighbourhood for the singularity (see Chapter ill, Example; .. , ,.,oreach U-mamfold W (e. g. by relabelhng or forgettmg some structure). 1 1(2)) The twisted theory is interesting in connection with represen.•. Then the theory VIU x M is defined by considering polyhedra P ting Z -homology (see §S). I .... n .f1tha two stage stratIfIcatIOn P::) S(P) and extra structure such that: j.
3.
£0
=
{p'l,
£1
= {xix ~ SOJ,
£n is all closed £n-l-
manifolds. This theory is 'ordinary' homology with coefficients obtain coefficients
Z one needs to orient the top stratum.
Zz'
To
We can think
of this theory as obtained from bordism by killing all manifolds except
i
1.
P - S(P) is a (V, q)-manifold.
1
2.
S(P) is a (U, q-n-l)-manifold.
i
3.
There is a regular neighbourhood N of S(P) in P and a
II
isomorphism
h: N ...•S(P) x C(M), which carries
S(P) by the identity
)0 S(P) x (cone pt. ). j ,~'
I 86
1, I
1f
I I
87
j
j
1 4.
h is an isomorphism
S(P) x (C(M) _ (cone pt.))
of (V, q)-manil'ulds off S(P) (where
is regarded as a V-manifold by part 3 of the
data).
I
bordism
1 WI
WI'
Then M x S(W1) is bordant to W by the V- bordism
- (nbhd. of S(W1)).
1 aL ==O. A V-manifold has no second stratum. P is then called a dosed (V/U x M, q)-manifold.
There is an
obvious notion of V/U x M-manifold with boundary and hence we have geometric homology and cohomology theories Notation.
j ~~
to W x {I}
There are long exact sequences
Form the product
B x C(M) and attach it
W==Wx
to a V-manifold.
{oj
X a ==O. If (W, S(W» is a V /U x M-manifold then S(W) x M bounds W - (nbhd. of S(W)).
L
X
B.
in W x I by the identity on S(W) x C(M). This constructs
aV/UxM-bordismof
a'
A) ...• Vq+n(X, A) ...• (V/U x M)q+n(X, A) ...• Vq_l (X, A). Ker
.•.• Vq(X,
Let !!!V, S(W)) be a V/U x M-manifold such that S(W)
L.
bounds the V-manifold
(V/U x M)* and (V/U x M
If U ==V then we collapse the notation to V/M.
Proposition 4. 1.
1m
X
elm
a.
If W is a V-manifold such that W x M bounds the
X + L + a q-l . V-manifold W', then we can glue W x C(M) to W' along W x M to ...•Vq(P, Q) .•.• Vq n(p, Q) .•.• (V/D x M)q n(p, Q) .•.•V (P , Q) . form a V /U x M-manifold with second stratum W.
in which ._... the homomorphisms ___ ~._~_.~ are defined as follows. -
X
is mUltiPlicationl --------1
by M followed by the identification of part 3 of the data .. L on Eepresentatives. a restr.!.c.ts to the see<.J~~~tratum~ e'
Corollar Y 4••3 Suppose V" IS a rlllg th eory an d [M] is not a is the iden~zero_divisor in V*(pt.) (i. e. ~ultiplication by [M] is injective) then
a(P:J S(P), f) ==(S(P), fIS(p)
etc.
---.- -----
l
1·1
(V/M)*(pt.) ~ ideal ge~::~~~~ by [M]
j Remark 4.2. jM.)
There is also a notion of killing a whole family
of elements simultaneously.
In this definition S(P) ==u (S(P).) i
1
1
I
i
Proof.
Consider the sequence with X ==pt. A ==0.
From now on we will, for notational simplicity,
and the neighbourhood of S(P). satisfies the conditions of the definition jhOmOIOgytheories. 1 homology theories. but with M. replacing M. This is then a generalisation of the killing 1
used in Chapter lIT to define coefficients. X
.•.•fB V
i
q
+ ...• (V /U x
q n
1q
+ ...•fB n
i
q-
of the Vniversal Co-
C. f. Remark 6. 1.
Proof of 4.1. formula.)
VI'"
The spaces
(Compare the proof of the universal coefficient X, A, P, Q play no role in the proof, so we
LX
== O.
.iofeach llltrlllslC stratum (see the last two sections).
manifold W x C(M). Ker
L C
88
1m X.
Let W be a V- manif old bordant to P by V /U x M-
By the universal
I
jcoefficient sequence, this theory is isomorphic to n~O(,)
@
Z[ t] the
iloca l'Isa t·lOn 0 f .'* nSO . at odd pnmes.
j
i
From results of Wall [14] we know that all the torsion in n~O
lis 2- torsion and hence that n~O (pt. ; Z[t]) 4
Iongenerators
Let W be a V-manifold then W x M bounds the V /U x M-
where p is a fixed
reSOIUti~n O.f ~[t], with a reduction to SO of the stable normal bundle
I
ignore them.
will hold for the co-
Z [t]) be the theory 'smooth bordism with Z[t]
coefficients' defined by considering p-manifolds,
a M.)
which the reader can check is a generalisation efficient Sequence.
Let n~O(,;
4. I becomes sequences like:
L
.•.•V
Exactly similar constructions
deal only with the
[M1], [M2]'
...
,and index(M.) ==0, i::; 1: ;
1
,
is a free polynomial algebra
moreover we can take index(M1) ==1
take M ==CP 1
2.
and to obtain index (M.)==O 1
jsubtract an appropriate number of copies of (CP )J.
t, I •
Now define theories
2
i ==1, 2, ...
as follows:
I
89
=
J1
nSo * ( " . Zll[2
J2
=
J1 /M , and inductively Ji 2
= Ji-l
/M .. 1
Finally let
Thus J is the geometric theory obtained from smooth bordism by introducing coefficients in Z[~] and then killing all the free generators
and then it follows from 4.4 that lj/(pt.) is an isomorphism since the
except CP.
class of CP
Note that by repeated use of 4.3 we have:
2
is the generator of both groups. k
2
tors by the formula lj/«M, f), qt ) Proposition 4.4. and is represented
---,------------
J*(pt.) ~ Z[~ ][t] where t has dimension 4,
geometrically by CP
2
labelled by l.
q
€
Z[t]
Now let K denote the theory ko*( ,) ® Z[~], i. e. the localiza-
V,
1/I«M x
f
0
11
1
= 1/I«M,
1)
),
f
2
There is a natural equivalence of theories
K'" J.
Proof.
Floyd [3], has constructed a natural transformation
Proposition 4. 7.
maps [Mn] to 0 if n
[W] =
Proof.
* 4k
and to index(M) t\
where
t is the generator of K*(pt. ) ~ Z[~][t], if n = 4k. He also proves that
L[W]
=
then
)
1
for brevity of notation).
CP
where
a.
1
n~O(pt.)
= 1
It follows that a.
-+
L is as in diagram
2
= M1 ,
M , ....
However,
2
1
1
0, and we can bord W to p' in the 2
Remark 4. 8.
1
1
=
M W.. 2
1
There is a similar geometric description of con-
nected KU-theory given by a similar construction using complex bordism
s n~O ® Zr~]
O.
theory J by using bordisms like C(M ) x W., where W.
acts on Z[t ][t] by
®1
=
is the coefficient of (M )n/4, since all the other M.
*
K*(pt.) ~ Z[t][t].
We will construct a natural transformation
and the Conner-Floyd map [3]. 1/1
in the commutative 5. MORE EXAMPLES
diagram Example 5.1.
Some theories which represent
---------~--
Let p be prime. links ~
90
This
L; a..W. in nS*O(,; Z[~]) where a.. € Z[~] and 1 1 1
1
1
n~O( ,) ® nSO Zr~ ][t] ~ K*( ,)
-+
11
0 if index(W)
W. are monomials in the generators
have index O.
n~O
n~O(pt.)
using the product formula for the index, we can read off index(W) as a.
s induces an isomorphism
=
€
4.6.
s : n~O( ,) ® Z[~] .•• K*( ,)
where n~O
We have to check that lj/
follows from:
Sullivan [10], using a method similar to Conner and
such that s(pt.)
0
* 4k,
is zero in J (here index(W) = 0 if n :
where
),
1
f), s(W)). I. e. that
(M x (W - index(W)(CP )n/\
lj/
11
0
2
The only non-trivial part is that if [W]
tion of real connected K-theory at odd primes. Theorem 4. 5.
k
(qM x (CP ) , f
and qM means M labelled by q.
is well-defined
-----
=
lj/ is defined on genera-
€
£, , (p)
a
€
£, , (p) * (p) 1
Z -homology.
Define a theory of singularities €
£, , ••. 2
where
p ----
by the basic
(p) is a set with p
91
points in it.
call an £-manifold a po-polyhedron. This is a ring
WI:-
theory (see §3), the ring closure of 'twisted Z -bordism' p
£
(Example
£
a n
=
{p'l,
£
= {p Ip
1
=
{(q)/q
even
is a closed £
n-
l and inductively
I-manifold with even Euler
3. 1(2)). An orientation for an n-dimensional p-polyhedron is a generator of H (P; Z ) ~ Z. p
n
Then an £-manifold is called an Euler space and can be thought of as a polyhedron with 'even local Euler characteristic'.
p
Let U be a connected ring theory with
space has Steifel homology classes,
U (pt.) ~ Z. Then U represents Z -homology if and only if o p ------p ---------Uq(L ) -+ Hq(L ; Z ) is onto, where L is the Lens space Sn/p of n n p ------n --------arbit~arily high dimension n. Now the generators
of H*(L ; Z ) are n
p
and f3, where
(l'
(l' E
p
Note that
and f3 have such structures. the freedom to 'twist'.
that the group used in the construction of {3 is Z for in the definition of a p-polyhedron. ring theory restriction,
The Casson-Quinn theories.
Finally we mention some examples of geometrically
p
not
The point is
L p as allowed
To make this restriction
into a
we impose the restriction that the group for the
normal block bundle of a stratum of codimension
l'
+ q in a stratum of
defined (co)-
homology theories, which do not fit as described into the pattern of this chapter. These are the theories whose coefficients are the surgery obstructions.
For details of the definition see Quinn [5]. A 'manifold' in space (corresponding
to fundamental group) and a boundary on which the problem is a homotopy equivalence.
Of particular
[10] that, at odd primes,
(U4G/PL)*,
interest is the theory
the Casson-Quinn theory corresponding to
The normal bundle of one stratum in the others can be 1. e. we can restrict
Example 5.3.
the theory is a surgery problem with a reference
on the normal bundles of the
E. g. impose stable orthogonal or unitary structures.
restricted.
using the fact that each stratum is even-dimensional).
1
We can modify T£ in various ways, still preserving its property
2.
(defined using the
see Halperin and Toledo [2]).
Euler space (Sullivan proves this by a careful induction on dimension 2
H
of representing Z -homology, for example:
(l'
16],
An Euler
The triangulation of a complex algebraic variety is an example of an
property follows.
Make stable restrictions
[12,
combinatorial definition of Whitney et al.,
is represented by the inclusion of L 2 in L , and f3 E H is represennn n 1 -+ L , where the disc is glued on by the p-fold cover. ted by L U D n-2 a n Note that the Bockstein of f3 is (l'. Both (l' and f3 are p-polyhedra and
strata.
Note that manifolds
are Euler spaces and that Euler spaces form a ring theory.
Proposition (see r 6]).
the representation
.
The theory of oriented p<polyhedra represents
p
Z -homology, in other words the natural maps T ( ) -+ H ( ; Z ) and p n n p Tq( ) -+ Hq( ; Z ) are onto. This follows from:
1.
characteristic}
1f
1
=
O.
which is
Sullivan has shown
this theory is isomorphic to K-theory,
as in
§4. This raises the question of whether there is a convenient geometrical 4
description for (U G/PL)*
(or even for G/PL*
itself) at all primes.
Also relevant here is the question of whether K-theory has a simpler geometrical representation
than that given in §4. Note always that, by
Chapter VII, all cohomology theories have some geometric representation.
Z rlJ Z n. Z .,. ~I Z (1' copies). p pup P Both these modifications are examples of restriction on the normal 6. COEFFICIENTS IN A GEOMETRIC THEORY block bundle system. For more information on the algebra behind pIn this section V denotes a general geometric theory, that is to polyhedra see Bullett [1]. codimension
l'
is the wreath product
say, a theory with singularities, Example 5. 2.
Euler spaces.
This theory was invented by Akin and Sullivan [16] and has interesting properties.
92
Define link classes by
labellings and generalised
orientations,
as in §§2, 3. We will explain how coefficients work for V-bordism.
We
leave the reader to take care of V-cobordism (V-mock bundles) and to formulate the appropriate Thorn isomorphism and duality theorems.
This
93
section is modelled on Chapter III, we follow the section headings of Chapter III, explaining where the difficulties lie.
Definition 6.2.
1
Iframing.
Let
is a bordism of M x
j
j[M] Let p be a short resolution of an abelian group G. A V-manifold
E
j
with coefficients in p can be defined exactly as in III §I and the theory,
denote the circle with the non-standard
V is a good theory if there is a bordism
lin V such that ~
Short resolutions
81
8
1
D of
81
to zero
to zero for each
V*(pt.). Remarks 6. 3.
1.
For ring theories our definition of a good
i
enjoys all the analogous properties.
In particular there is a universal
coefficient theorem.
,;theory coincides with Hilton's, see r 4; 1. 9]. For general theories Hilton's 1 definition is equivalent to insisting that
[each V-mock bundle T/. Remark 6. 1.
Coefficients
p
is an example of killing, as des-
cribed in §4. To make the notation fit with §4, let V = V ® F I
(Va The trans-
manifolds labelled by elements of B ) and U = V I8l F. a 1 1 formation U x L(r, p) ...• V is given by ignoring the label on the first 1
factor.
1
Then (V, p)-theory is the theory obtained from V
simultaneously the elements
{L(r, p)lr
E
B
1
1
by killing
1.
I"
4(1),
w:W~'
;t
jOf L(b 2'
p)
-
81
is cobordant to zero for
This is in fact sufficient to prove Theorem
:ota::~d:::~
thenwe eaneompletethe eon,truetton
plug in the bordism
1(inmost cases
x
T/
2
D= D
D of
81
to zero wherever appropriate
and the construction coincides with the old one).
,I
jFunctoriality
Longer resolutions
The best result we have, for a general theory, is the following:
The description of coefficients in resolutions of length :s 4 in Chapter III is again an example of killing (made precise as in 6. 1 above). To make this work for a general theory we need to regard
L(b., p) x M 1,
Theorem 6. 4.
Coefficients in a short resolution of an abelian
I
igroup gives a notion of coefficients which is functorial ','(I) for good th eones .
as a (V, p)-manifold for each V-manifold M and each link L(b.,1 pl. NOW,l(2) on the categ ory 0f d'lrec t sums 0f free a be l'Ian groups an d 0dd t orSlOn . i each stratum of L(b., p) is a disc, so we can regard L(b., p) x M as i ---. '---------,-...-----1 1 igroups. a stratified set with each stratum a V-manifold, and the only possible problem comes from 'restrictions
on the normal bundle'.
In general
Remarks 6. 5.
The universal coefficient sequence is natural
this problem is solved by endowing L(b., p) with the universal restric-
!(andhence, usually, splits) in exactly the same cases.
tion, namely framings of each stratum which fit together in a standard
[
1
way (i. e.
L(b., p) is an object in the theory of framed manifolds with)
coefficients - ~table homotopy with coefficients).
Proof of 6.4.
1.
Exactly the same proof as III §4, using
For the Q-stratum there IRemark 6. 3(2). 1
is no problem (framing is equivalent to orientation).
For the I-stratum'
we have to frame each I-disc extending given framings near the ends. Orientation considerations
94
We now make some more precise
Coefficients are always functorial on the category
This is seen as follows.
Define 0' ( ; F), where
,abelian group, by allowing no singularities
the non-uniqueness of framings of circles implies:labellings
that the framing may not be possible. statements.ialways
Finally
Step 1.
:offree abelian groups.
imply that this is possible but there is non-
uniqueness - there are two possible choices for each I-disc. for the 2-stratum,
2.
q
F is a free
in the representatives
only) and codimension 1 singularities
(i. e.
only in the bordisms
:(this requires only the definition of O-links, which, as seen earlier, holds). Now {2' ( ; F) is the same as n ( ) ® F by exactly the . q q jargument of ill ~4, but with all levels of singularities reduced one step. ,
95
We leave the reader to check the details here; geometrically all that is
REFERENCES
FOR CHAPTER IV
required is a construction (not unique) of I-links, which we gave above. Step 2.
Coefficients are functorial for odd torsion groups. The idea here is to use the fact that 81 has order two to complete
the construction of the 2-links for a 3-canonical resolution. -1
stage we have S
labelled by g
E
-1
S. Bullett.
Ph. D. thesis,
[2]
S. Halperin
and D. Toledo.
Ann. of Math. [3]
G and g has order t, t odd. We
have to plug in a bordism of g81 to zero.
!.; 1
Take
copies of Sl x I -1
framed so S
At the final
[1]
is at both ends and glue all the copies of S
[4]
required bordism. g + g + ... Step 3.
The new singularity is labelled by the relation
[5]
+ g (t times).
@
group. Define SV'(;
[6]
G1, where F is free and G1 is an odd torsion
P1
G) by using the two definitions given above.
q
-
-
Precisely a generator of il" is the union of a generator of il ( ; G ). q
A 'bordism' is similarly a union of bordisms. and h: G - G' a homomorphism. (II
F
••
Now let G'
=
q
F'
G'
1
[91
F'
}
[10]
•. G'
since there is no non-trivial
homomorphism
that we can define
of relabelling
of too high a codimension.
Similarly
is functorial.
Steps 1 and 2.
1
-
C. P. Rourke.
Representing
Soc.,
257- 60.
case.
as Theorem
96
of universal
Algebraic
5 (1973),
Topology,
of surgery.
homology
C. P. Rourke and B. J. Sanderson.
coVol.
I,
Ph. D. Thesis,
87, (1968),
D. A. Stone.
classes.
Bull.
Block bundles:
Lon. Math.
Ill.
Ann. of
431-83.
C. P. Rourke and B. J. Sanderson. Berlin
Introduction
to pi topology.
(1972).
Stratified
polyhedra.
Springer-Verlag
Geometric
topology
lecture
topology
seminar
lecture
D. Sullivan.
Geometric
D. Sullivan.
Combinatorial
relabelling
and
can be relabelled.
[13] [14]
Thus
group follows
from
R. E. Strong.
[15] [16 ]
or coefficients
are functorial
Notes on cobordism
C. T. C. Wall. Math.,
Determination
72 (1960),
N. A. Baas. Math. Scand.,
of Chapter III goes through with obvious changes The constructions
invariants
notes. notes.
of analytic
M. I. T. (1970). Princeton spaces.
(1967). (To
appear. )
with singularities
to the correct
D. Sullivan.
E. Akin. Amer.
general
cobordism
No. 28.
(1969).
[11]
the proof.
Products, Bockstein sequence and rings. The rest
On the splitting
formulation
[12] F'.
an element
bordisms
That it is isomorphic This completes
G
h[M] by simply
we never meet the problem
il"
between
Notes
notes No. 252.
1
This means
Lecture
Aarhus Univ.,
A geometric
Springer- Verlag,
\B 1
[8]
Then h splits as
@~
G
F. S. Quinn.
Math.,
1
@
classes.
180- 201.
Princeton
Coefficients are functorial on the category of direct sums. Let G = _F
sequences.
Whitney homology
The relation
Springer-Verlag
P. J. Hilton and A. Deleanu.
(1970),
(1973).
511-25.
P. E. Conner and L. E. Floyd.
efficient
1
Finally glue on one copy of 8 x I by one end. This constructs the
Stiefel
96 (3), (1972),
and K-theory.
together.
Warwick University
Princeton
of the cobordism
U. P. (1968).
ring.
Ann. of
292-311.
On bordism 33 (1973),
Stiefel
theory.
theory of manifolds 279-302.
Whitney homology
Math. Soc.,
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205 (1975),
classes
and bordism.
Trans.
341-59.
in the
under the same conditions
6. 4.
97
Proposition 1. 1.
V"Equivariant theories and operations
jsuch that
IKI = X
Suppose G acts on X.
Then there exists
and there is induced a good action on K.
1 .;
!
Proof.
It follows easily from definitions that X/G has a pl
jstructure so that the quotient map q : X - X/G is pl. 11LI =X/G, In §l of this chapter we describe a further extension of mock bundles, to the equivariant case - the theory dual to equivariant bordism classes.
The remainder of the paper is concerned
with the case of Z - operations on pl cobordism.
In §3 we expound the
2
'expanded squares'
('expanded' rather than the familiar 'reduced' because
of our indexing convention for cohomology) and in §4 we give the relation with tom Dieck's operations [5].
§5 describes the characteristic
classes
so that for each g EG thesubpolyhedron
Choose q{xlgx=x}
jis a subcomplex of L. Then define a EK if and only if q(a) E L then
-IK
is the desired complex.
and in §2 give a general construction which includes power operations and characteristic
K
A mock bundle ~IK jG on E(~)
is a G-mock bundle if there is an action of
which induces an action on K. Let T~(K) denote the group
jofG-cobordism classes of G-mock bundles over K. If the action on
IE, W
is good (i. e. if g{3a=
p a implies g I (3a = id. ) then the subdivision
iprocess can be carried through equivariantly simply by subdividing all lblocks in an orbit isomorphically.
associated to Z - block bundles and in §6 we give a result inspired by 2
It follows from Proposition 1. 1 and the subdivision construction
Quillen [3] which relates the total square of a mock bundle with the
1
transfer of the euler class of its twisted normal bundle.
!that there is a G-homotopy functor T~ on G-polyhedra defined by con-
This leads, in
some cases, to the familiar connection between characteristic and squares.
classes
!sidering only good actions.
Finally in §7 we give an alternative definition of squares,
based on transversatility.
Proposition 1. 2.
This is like the 'internal' definition of the cup
In fact we have: There is a natural isomorphism
product (see II end of §4). 1. EQUIVARIANT MOCK BUNDLES
Proof.
Let G be a finite group and X a polyhedron. X we mean a (pl) (i)
!Then q* ~ has a natural G-action and we can define
if e EGis
If X, Yare
2
1
2
1
G polyhedra then a map f: X - Y is a G-map if
From 1. 2 it is easily seen that the work of II carries
below how to define equivariant cobordism via G-mock bundles. Then we say G acts on K
provided for each a E K and g EG, ga EK. The action is good if in addition whenever ga
98
=
n
over to the
icapproducts and Thorn and Poincare duality isomorphisms. If U is a geometric theory then there is also a notion of equi-
bordism of X by equivariantly mapping G-manifolds into X and we show!variant U-bordism and equivariant U-mock bundles.
I I.
[q*
:equivariant case when the action is free, for example there are cup and
We then have the concept of equivariant'
Suppose now G acts on X = K
=
2
the identity then ex = x for all x EX.
f commutes with the G-action.
e[ ~IL]
!g is easily proved to be an isomorphism.
for all g , g E G and x EX, g (g x) = (g g )x. 1
(ii)
By a G-action on
map G x X - X satisfying
Let [UL] E Tq(X/G), with L as in the proof of 1. 1.
I
We omit details.
We shall see in the next section that the case of a G-mock bundle
i~1K with the action in K not good is extremely interesting
as the power
;operations spring from consider such cases.
a we have g/a = id.
99
i ,~
'1
2. THE GENERAL CONSTRUCTIONAND THE POWER OPERATIONS
suppose given a non-trivial i
Let W be a free G-polyhedron and JaG-complex point polyhedron
F c IJ I
=
j8 : Zq(K) -
with fixed
:
X.
Let U be a geometric theory.
zcg(~),
homomorphism
J.l : G -
where Z denotes isomorphism bundles,
2,.r
Define
classes
of U-mock
--------r
by sW = ~ x ... x ~ with G action given by permuting factors via J.l. I 1S commutes with ~ubdivision of K and can be seen to define external_
We have the following commuta-
tive diagram of homomorphisms:
;power operations qr
PI (J.l, W) : Vq(X) _U
P (J.l, W) : Uq(X) _Vqr(X o
(2. 1)
lby P.(J.l, W)
=
<1>.
1
J
0
1
linternal operations
Here action.
1 (~) =
e- 1 (~
r* is restriction
bundle with fibre
r
(X
X
G
W)
x W/G)
s.
When 11, is defined (see above) then we have . q qr+n p! (J.l, W) = ! s : V (X) - U (X). 0
13. THE EXPANDED SQUARES
x lW) and G acts on X x W by the diagonal I,
and 11! is composition with the trivial mock
Now restrict
W/G, when this is defined, that is, when M x W/G
is a V-manifold for each V-manifold
attention to ordinary
iT*( , ) as usual) and, in the construction :let G = Z
M, (e. g. if W/G is aU-manifold
122
and V is a ring theory).
=~
pi
cobordism (denoted
of the end of the last section,
and let W = Sn, the pi
n-sphere
with antipodal action.
:We obtain the external and internal expanded squares: i
The whole diagram is natural for subdivisions of G-bundles over J'
1
for G inclusions
J
C J and W o 0 the free G-cobordism class of W.
C
W. Further
,
There is a relative version got by replacing
•
I
depends only on
J, Q, and F by
(J, J ), (Q, Q ), and (F, F ) respectively. 000
The construction of can be made for more conventional o 1 types of bundles, for example vector bundles, spherical fibrations. Example 2.1. let W be the sphere
Let u;X be a G-vector bundle with G = Z Sn with antipodal action.
where G acts trivially on the fibres of u
1
Then u IF
=
U
@
2
o 1 and antipodally in the fibres
of u. It easily follows that (u) = 1I*(u ) ® 1I*(l ) + 11*(U) where o 0 10 2n 11 is the canonical line bundle on P and 11 and 11 are the obvious n
1
and
u
2
projections.
l
'for n i
=
0, 1, 2, ...
i
Remark 3. 1. The name I expanded square I gains more credence i •• 2r+n n ifrom the observation that we can choose representatives ~ for iSq~+rr {l so that I
•
Power operations Now let L;
100
r
denote the symmetric
group on r symbols and
;and from definitions we have [~2r]
= [{l
u
[{l. A similar remark 101
applies to
Sq~rn
=
i
I
0, 1.
The following lemma is easily proved. Lemma 3.2.
(a)
i*Sqn[~] =
(b)
1
= [{]
Sq~l{]
r~]x [n
where --
u [{]. i : X x X x pt. -Xxxx
Snis; Z2
the inclusion induced by the inclusion of a point in Sn. (c)
,weget sq~([~]
Let ulK be a block bundle and let tu be its (canonical)
-------
1
X
Xz
=
[M x M
2
(d)
sq~+m[Mm]
sq~[~]
U .Sq~[7]]..
is illuminated
U 1/)
Jbythe following.
-------------
Xz Sn -X x
=
k
Thorn class then Sqn(tu) is the (canonical) Thom class of E(u) x E(u)
U [1/))
The relation between Sq~~ u Sq~TJ and Sq! (~
Sn. 2
Corollary 3. 6.
•
z2
X
There is a commutative diagram
Sn] wher~ M is a closed mani- )
fold here regarded as an m-mock bundle over a point. Proposition Proof.
n n n+r . Sq , Sq and Sq, _ar_e_h_om_o .._m._o_r_p_hl_s_m_s_. I
3. 3.
1
0
.
It is sufficient to show Sqn is a homomorphism. 1
I
AbUSingj
the notation we have
! m !i
which the square is a pull back. Proof.
The last factor is a composition (of mock bundles)
This follows from 3. 5 and the diagram
~ x 7] x Sn - K x K x Sn - (K x K x Sn)jZ 2' but the O-mock bundle K x K x Sn - (K x K x Sn);Z o o E T «K x K x Sn);Z ). The result follows.
gives the class 2
2
The following is immediate from definitions. ~ which each square is a pull back. Proposition
3.4.
Let i: «K x L) x (K x L) x Sn);Z -
- (K x K x Sn)jZ
2
x (L x L x Sn)IZ
2
r
Proposition 3. 7.
1
j
2
be defined by' i [x 0 , x 1 , y 0 , y 1 , z] = ([xo' Yo' z], [x 1 , y 1 , z)). -----, sq~r~ x 7]] = i*(Sq~[~] x Sq~[7])) for any ~, 7].
Then
Proof.
sqn1 _and Sqn0 __ are ring homomorphisms. ~~ ~ __
0
0
0
n
where
=
(x, y, z).
j
i
1
:1·:
Proof.
This follows from commutativity in AXA
(Qx P n ) x P m --+- «Q x Q x Sn)jZ 2 x (Q x Q x Sn)jZ
Ii
I
It is sufficient to show that Sqn is a ring homomorphism, 1 but from the formula in 3.4 and the commutative diagram . , 102
0
Q x P n x Pm - Q x Pm x P n is given by j{x, y, z)
,
Corollary 3. 5.
j*SqnSqm[~] = Sqmsqn[
I
I
I
IIS~ (Q x P
m
x Sm x Sm)jZ 2
IIS!k
j
)x P
2
AXA n
~ «Q x Q x Sm)jZ x (Q x Q x Sm)jZ x Sn x Sn)jZ 2
2
103
2
where k shuffles the spheres and each
A
is a suitable diagonal map.
Proposition 4. 2. (ii)
In [5] tom Dieck defines operations in the smooth case analogous and Sq.
=
i
Sq IJW, where on the ri~ht we
have the usual Steenrod operation.
4. RELATIONS WITH TOM DIECK'S OPERATIONS
to our Sq
i
IJSq!W
(i)
IJSq~W
(i) follows from the axiomatic description of Sqi and
Proof.
That the definitions agree in the smooth case
= IJRiW.
o 1 from (ii), which comes from the fact that follows from 3.2. c. We now look at the relationship between our internal operation Sq~ and tom Dieck's internal operation. isomorphism,
By virtue of the Tholl1
IJ[P .] n-l
see e. g. [1], one readily proves
=
0 unless
n
=
i.
5. CHARACTERISTIC CLASSES T*(X x P ) n
S'"
T*(X) @ T*(P ) n
In this section we present a special case of the construction of §1.
and
Let u/K be a block bundle with involution f: E(u) - E(u) satisfying fp-la
=
p-la
=
and IK/
{x : f(x)
= x l.
Recall that the inclusion
X = I K ICE (u) is the projection of the Thorn class tu and by virtue of where x is the euler class of the canonical line bundle l;P and by n n the involution we have [tu] E T"Gs(E(U),E(u)), G = Z2' Now we may direct construction we have p : P 1 - P , the usual inclusion, and the . x nn apply the construction of §1 to get projection of Xl is the inclusion P ....• P. Thus n-l n Sqn(~q) = ~ Rq+i~q @ xi o \ i=O and q +. \ and the R 1 are tom Dieck's internal operations, with a change of sign in the indexing.
W
-n
(u)
=
[tu] E T-s(X 0
,
W' (u) = n-s
[tu] .
<1>,
I
Proposition 4. ~• Proof.
Sq~+qw
.
= ~
[P
.]R-q-i~. i=O n-l
p, T*(X) ® T*(P ) ...•T*(X x P ) -+ T*(X). n
n
T
n-s
n
(X). ,
W· (u), w· +1(u), ... , W· + (u), ... -s -s -s n teristic classes of (u, f).
are the Z
The classes
Denote by u
Consider
I
E
x P ),
@
l
n
the block bundle E(u) x z
charac-
2
Sn - X x P n'
In
2
the case that u is a vector bundle with antipodal action then u ® l
n coincides with the usual tensor product with the canonical line bundle.
From definitions we have: Then PI xi has projection X x P ....• X and therefore PI xi = [P .]. 1. . n-1 .' n-1 Further if UX has projection p t : E(~) ...•X then ~ ® Xl has projec., Proposition 5. 1. W (u) is the euler class e(u ® l ), and -n ------n-tion E(~) x P . - X x P and composing with X x P - X we see that I s i n-l n n W· (u) = e(u). PI ~ ® x = [P .H. The result follows. -s . n-1 To get some geometric insight into the meaning of the Now let IJ: T*(-) - H*(-; Z ) be the Steenrod map (the identity 2 i consider the diagram on representatives, see IV 3. 1(3)). Let Sq be the usual Steenrod squarE (with a change of sign),
104
105
Let uIX
E(W (u))-
-n
t
C
Xl ::
+
2
to be the complementary block bundle.
n + uniqueness of u follows from 5. 1 of [4].
I I
where p is the projection of X x P C (E(u) x Sn)/Z n
2
1
(tu) after subdivision so that
appears as a subcomplex.
we see that we may regard
From the diagram
E(W (u)), after dividing out the Z
-n
2
action,
as f-l(X) where f is an equivariant approximation to X x Sn-XCE(u) which is transverse
a
-1
E
theorem and an induction over the cells of K x Sn,
where Sn has a suitable equivariant cell structure. native definition of the characteristic
This gives an alter-
0
K, P~ (a)
= pu-1 (a) 0
n E ~ and further suppose given a pi -1 -1
G
bundle v IE ~ for the inclusion so that Pv p ~ (a)
= Pu-1 (a)
m v .
given by p(x, y) = (p~(x), y). We need the following generalisation
Let gS;K be the trivial block bundle then
0
formula,
of the clean intersection
3. 3 of [3]. and isomor-
O
map
Y
-11 K I = P(v),
t
into E(u) we get an equivariant transverse . f : K x Sr+s - E(u) WIth f
-1
q Theorem 6.1. Suppose ~qIX is in u +m with normal bundle + n + Then e(u )SqoW =p!e(v ), where p: E~ x Pn ..•.X x Pn is
W' (gS) is the class of the projection K x P - K. Lemma 6. 2. Suppose given a block bundle w IX l' -------~-~-l' Now suppose vr+ 1;K and uS;K are block bundles with involution phism w @w !Y ~ w Iy, y C X. Then o 1 h -an dthtth a ere IS an equlVarlan t" Isomorp hIsm uSI1> '" vr+l~ = gr+s+l were i (i*W u e(w )) = j*j, W for ~ E Tq(x) g is the trivial block bundle with standard involution and the involution on I! 0 1 . s r+ 1 " . s r+ 1 I u @V IS mduced from u @v ~ p*v E(u). Then we have where u K x gr+s C E(gr+s+ 1) and composing isomorphisms and the projection i 0
normal
n Pv (E ~). In
classes.
From the alternative definition we have
°
0
this situation we say ~ is in u with normal bundle v.
to X. Such a map f may be produced by using the
Proposition 5. 2.
The existence and
Now suppose given a mock bundle ~q;K with K = X and q m E ~ C Eu where u is a pi bundle with fibre R + and for each
C
transversatility
bundle with fibre some euclidean space, and
n
u IX x P
x
pi
be a pi
suppose A: X x P ..•.(X x X x Sn)/Z is given by A(x, [y])=[ (x, x, y)]. n 2 Then there is a bundle inclusion u x P - A*(u x u x Sn/Z). Define
2
)0
E(w)' 1
i
X
where P(v) denotes the mock bundle E(v)/Z
1
~ jl )0
E(w)'
- K. We have then: is the diagram of inclusion, and E(u)' denotes the pair (E(u), E(u)).
Proposition 5. 3. P(v) •.•.
IKI.
In the above situation, W' (us) is the class of --------~
l'
------
Proof.
Consider the diagram of inclusions
6. SQUARESAND EULER CLASSES The purpose of this section is to prove Theorem 6. 1 below and derive consequences.
The result was inspired by Theorem 3. 12 of
Quillen [3]. 106
107
p :E
~
x P -X n
x P
n
and
p': (E x E x Sn)jZ -(XXXXSn)!Z ~ ~ 2
be the projections. Now apply 6.2 to the top square to get
Now from definitions i!p! = C*i! and so we have Since i = 1
l l
we have j*i,W =
0
l *l
o.
1 •
~;i; 1 = ...
*j,W. Now apply (II;2.4) to the
square,
i, PI e(v +) .... (1)
Now apply 6.2 to the element p~(l) and the bottom square to get
and get
since from definitions ~*p; (1)= SqnW• Since i;P; = c'*i; and x-. a ... + ~ c=c'~u we have from (1)and (2)that iIP,e(v+)=j,(sqnWue(u)) v •. a and hence the result since iI is the Thorn isomorphism. Suppose now that u, v are vector bundles. Then u, v have underlying pl structure (see [2]),and itis easy to see that u+ = u ~ l + n and v = v ~ l (see for example [1;p. 138]). From 5. 1 we now have
• X
Y
Again apply (II;2.4) to the square
i i*W.
l *l
o I!
0
1
l E(w )'
r·
10
1
i
1
Y
•. E(w
•
o
@w)'
1
t
n
the following corollary.
lo
E(w)'
Corollary 6.3. If ~ is in u with normal bundle v and u, v admit vector bundle structures then
1
and get i i*i i*W, which is i (i*W u e(w )) by (II;2.6). I! 0 11 0 01 Proof of 6.l. E~
xp
Consider the diagram ~~ n i'
ni E(E
~
x P )'
v
xxp
where
x E
E (E v
tV
x Sn), jZ
2
Proof.
C'
t u
v
n
c
E(E
Corollary 6.4. For pl bundle u/x with (canonical) Thorn class t , Sqn(t ) = i,e(u+), where i: X x P -+ X x E(u) is the --u au. -n -inclusion.
n
~
x.P)'
XCE u
'n uj'
__________
u
x E
u
x Sn),
~
x
u
-+
-+
X by
E . u
7. THE TRANSVERSALITY
..••. _ (X x X x Sn)jZ 2
Let
jZ
Apply 6.2 after replacing E ~ C Eu
2
U j'
n
c, c' are collapsing maps.
E (E
id
DEFINITION
OF THE EXPANDED
SQUARE
The previous section raises several interesting questions, e.g. the relation between u+ and u ~ l in case u is a pl bundle with n Z -action. This section clarifiesthe situation, see e.g. Proposition 7.2 2
108
109
2
below.
Proof.
This is achieved by a further definition of sq~ as a special case
of a general construction which we now describe.
-
is self transverse
Let c : X - X be an r-fold covering map. We define a function : Tq(X) - Trq(X) as follows. Let w t'X be the vector bundle with c c fibre at x E X the vector space with basis set c -1 (x). Now let E
Tq(X) then for some large
N we may assume
~ in eN.
Then the quotient map «X x RN) x (XXRN)X Sn);Z2
x RN) x E~) x S~-
with self intersection2:\E ~ x E ~ x Sn);Z2' which gives
Sq~~ after restriction
P
[~]
Consider
f: (E ~ x (X x RN~x
to X x P.
n
1.
But the restriction
f to X x P n is E(vc ) and its self intersection
of the domain of
is P c (1P
~ is in the trivial
x <,. t)
The
n
result follows from the commutativity of the subdivision with the operaDefine ~c in e ® wc by tion of making self transverse. E(~ ) = {y ® z : y E E(~), c(z) = Pt(Y) l. Observe that ~ = c, c*~. Now We shall use the new definition to prove: c 1. <, C define a (vector) bundle v /E(~ ) by taking the fibre over Y ® z to be c c Proposition 7.2. Suppose that u is a block bundle with Z -action. the subspace of the fibre of e ® wc at p ~(y) generated by vectors a ® b,
bundle eN with normal bundle v, say.
0
n
with c(b) = P t(Y) and b *- z.
Then Sqotu
Further define f: E(V1.)- E(e ® w ) by <, c c f(y ® z, ~a. ® bo) = Y ® z + ~ao ® bo. Note that the image of f is a 1
1
1
closed subspace of E(e ® w).
1
Now suppose
~ has base
K,
so
Over a vertex of K, E(v ) falls into r-distinct pieces and under f these c pieces intersect in the fibre of € ® w over the vertex. Shift f so c
the r-pieces
and all the intersections
are pairwise transverse.
This can
is the block of P c (~c ) This is the start of an inductive process which we call
'making f self-transverse'.
The resulting self-intersection
is the inclusion •
It follows from 7.1 that Sqntu can be obtained by making
n
X x S - E(u) x P
n
self transverse.
0
On the other hand e(u ® l ) is
n
obtained by making X x P n x SO- X x P n - E(u ® l n ) self transverse • Both E(u) x P nand E(u ® l n) are quotients of E(u) x Sn under a (free) Inductively make X x Sn - E(u) x Sn equi-
variantly (with respect to the diagonal action on the right) transverse I' : f
-1
n
(X x S );Z
2
- Xx P f
n
is the projection of (u ® l ) = W (u). On n
the other hand X x Sn - E(u) x Sn - E(u) x P restricting
f and intersecting one with the other which would give the cup product
[f] u (f] where f is regarded as a mock bundle over E(e ® Wc ). It is not hard to show that the operation is well defined and functor i- then is just the
--
n
-n
is self transverse
Corollary 7. 3.
If u is a block bundle which admits a Z - action -
--------------
2
(i)
Wn(u) ~s independent of the choice of Z 2- action, and
(ii)
!! u
The description of P (~) considerably simplifies in case c p ~ : E ~ - X is an embedding. In this case one can inductively make
e(u + )
=
p : E (c, c*~) - X self transverse. As an example consider the double cover °c : Sn _ P and p t : P 1 _ P the inclusion so that [~] E T-1 (P ) is the generator.
n
<,
n-
n
1
Then P (~) = 0 since Sn- _ P c
n-l
- P
n
n
can be shifted
to have empty self intersection. Theorem 7. 1. n
then Sq ~ = P (lp -0 c
110
and
to the intersection gives f' again.
r-fold product (see II end of §4),
c
to
X x Sn. Suppose the result is f: X x Sn - E(u) x Sn. Then
is the total
space of P (~). The self-intersection should not be confused with the c result of intersecting f with itself, which is got by taking two copies of
al for bundle maps, and for the trivial r-fold cover P
2---
i,,-n W (u), where i: X x P nun - E x P ~
Z2-action, by definition.
be done using (TI;4. 1). The total intersection over the vertex.
Proof.
IKI =X.
1. C
=
Proof.
reduces to a pl
---
_b_un_d_l_e_t_h_en_
e(u ® l ) = W (u). n
-n
(i) is immediate and (ii) follows from 6.4 and 7.2.
Let Xm be the class of the inclusion PIC P and let mm H(m-l, n) C P m-l x P n C Pm x P n be defined by {(x, y) : LXiYi=O }, see e. g. [1] p. 164.
the double cover, Let c : Sn x X - P n x X be --------
x ~). n 111
Corollary 7. 4. H(n, m)
C
P
n
x P
Proof.
Sqn(X ) is the class of the inclusion o m ------------
VI·Sheaves
• m
X
m
is essentially the Thorn class
Sqn(X ) = i, e(l 1 i8ll ) but this is H(m-l, n) om. mn construction, see Theorem 6.6 of [1], p. 164.
C
t
l
P
.
So
m-l m
x P
n
by direct In this chapter we extend the treatment of coefficients in Chapter
Definition 7.5.
For any block, vector or pl bundle u define
W (u) by i, W (u) = Sqn(tu), where i: X x P - E x P . -n ,-n 0 nun In view of 7. 2 no confusion can arise from this definition.
III to cover sheaves of abelian groups.
We work always with pl
co-
bordism but everything that we say can be extended to an arbitrary under the conditions of IV 6.4.
theory
§§4 and 5 in fact extend unconditionally.
The general definition of sheaves of coefficients does not have all the best REFERENCES FOR CHAPTER V [1]
properties one would hope for and we will explain where the difficulties
T. Brocker and T. tom Dieck. Kobordismen Theorie.
Springer-
In §l we recall the basic properties of stacks and sheaves and in
Verlag lecture notes no. 178. [2]
R. Lashof and M. Rothenberg.
Microbundles and smoothing.
D. Quillen.
Elementary proofs of some results of cobordism
theory using Steenrod operations.
Advances in Math., 7 (1971),
C. P. Rourke and B. J. Sanderson. versality.
[5]
Annals of Maths.,
T. tom Dieck.
The
The main theorem asserts that, if the stack is 'nice',
Block bundles ll: trans-
87 (1968), 255- 77.
Steenrod-operationen in Kobordismen-Theorien.
Math. Z., 107 (1968), 380-401.
then there is a
spectral sequence expressing the relation between simplicial cohomology and cobordism with coefficients in the stack.
29-56. [4]
§2we define the theory of mock bundles with coefficients in a stack. definition is functorial on the category of all stacks of abelian groups.
Topology, 3 (1965), 357-88. [3]
lie at the start of §4.
In §3 cobordism with co-
efficients in a sheaf is defined by means of a simplicial analogue of the Cech procedure.
In §4 we discuss an extension of the methods used in
the previous sections and give an example of 'Poincare duality' between bordism and cobordism with coefficients in the sheaf of local homology of a Zn-manifold.
Finally in §5 we extend the methods further and give
examples which suggest the existence of a bordism version of the Zeeman duality spectral sequence [1]. 1. STACKS A stack of abelian groups over a cell complex K is a covariant functor T: ~ -
A homomorphism between stacks,
natural transformation of functors 1/>:
T -
T'.
over K will be denoted by S IK or simply S.
112
T
IK,
T'
IK, is a
The category of stacks If
T
is a stack over K
113
and K'
C
a u' ,
1
C
a
l'
1
<
=
a')
2
u' Ca. 2
T(a
If
2
1
T
over K' is the stack
<
a)·
T/K
a'
2'
T(a < u ), a , a 1
2
1
E 2
1
K. If
xI
a' E K' 2
"1'
a
2
a
a
2
EK
Ie.
E
J. (1IJ)(a ,
< 12
a )
XI)=(TxI)(u
1
is a stack, and J
C
T(a
< 1212
IKI
p~ : E -
(a)
(T x I)(a x 1) = (T X I)(a) = T(a)
(b)
E
J.
I K I = X,
E
K, p~l(a) is the interior of a (T(a), q+dima)-
E
K, 1f
uaY
for each a
r(a. < a)p~l(a.) ~
[a : a]T(a < a)p~ 1 (a ), where
i
i
i
is the image of the T(a.)-manifold [bordism] under the
1
1
< 1
relabelling morphism
Let X be a polyhedron and F a presheaf of abelian groups over
~q/K consists of a projection
1 1
a ), a , a
such that for each a
manifold [bordism],
K, the restriction
I
=
A (T, q)-cocycl~obordism]
'
TI.T is defined to be the stack over J, such that (T J)( a) = T( a), a
Let T be a stack of abelian groups over an oriented cell complex
/K' such that
is a stack, T x I will denote the stack
over the cell complex K x I, such that: for every aEK- , (TxI)(a
(r' l'
,
T'
T(a.
a); [a. : a] is the incidence number and its 1
effect is a change of orientation iff [a. : a] = -1. 1
uaY - 0 uaY is called the block K)::> st(a2, K», a, aI' a2 EK. ~ver a; (T, q)-cocyc1es [cobordisms] ~, T/ over K are isomorphic, K l 2 We briefly recall the notion of simplicial cohomology with coefwritten ~ 2" T/, if there is a (Pi) homeomorphism h: E ~ - E T/ such ficients in a stack and its relation with Cech cohomology. Let T/K be chat h I ~(a) is an isomorphism of T(a)-manifolds [bordisms] between a stack over the oriented simplicial complex K. A (-p)-cochain, fP, ((a) and 1] (a). Suppose given ~q/K and L C K, then the restriction
X. If K is a cell complex, FK(a) = F(st(a,
K», F (a
with coefficients in ~
E
T,
F induces a stack, FK/K, by:
K, an element of T(uP); (-p)-cochains form an abelian group by
coordinate addition, C-P(K, T). There is a coboundary homomorphism: oPfP(JJ+1) =
L:
+1[a:
~(a) =
< ( ) = F(st(a1,
is a map which assigns to each p-simplex
o-P : C-P(K, T)- C-p-l(K,
The manifold [bordism]
T), given by
uP+1]T(a<
~+1)fP(a).
\ IL
is defined in the obvious way and it is a (T I L, q)-cocycle [cobordism]
over L. A (T, q)-cocycle [cobordism] over (K, L) (L
chain comPle:=:'~ its cohomology is by definition HP(K, T). If F IX is a sheaf and FK the induced stack over K, let ~
K) is a (T, q)-
cocycle [cobordism] over K, which is empty over L; - ~ is the cocycle cobordism] obtained from
{CP(K, T), oP} is a
C
~ by reversing the orientation in each block.
If ~o and ~1 are (1, q)- cocyc les over (K, L), then ~0 is cobordant to ~--~ \ if there exists a (T x I, q)-cobordism T/ /(K x I, L x I) such that 1
~I K xli}
.
= (_1)1~. (i = 0, 1). Cobordism is an equivalence relation 1
denote the covering of X formed by the stars of the vertices of K. The
Ind we define Uq(K, L; T) to be the set of cobordism classes of (T, q)-
nerve of ~
:ocycles over (K, L); Uq(K, L, T) is an abelian group under the opera-
is known to be equ~ to K and this fact induces a canonical
{CP(K, F ), oP} .:: {CP(~, F), 6} of chain complexes; K •.. stands for ,eech'. Therefore, because the coverings by stars are a
:ion of disjoint union; we call it the q-th oriented (Pi) cobordism group
cofinal system in the system of all coverings of X, we get
nation' homomorphism
identification
)f (K, L) with coefficients in
T.
If T' <J
T
then there exists an 'amalga-
am: Uq(K', T') - Uq(K, T) like in the case of
)rdinary mock bundles (see Chapter II). In the proof of the subdivision limHP(K; F ) = limHP('U, F) = HP(X; F). K
-+K
-'U
2. COBORDISMWITH COEFFICIENTS IN A STACK Throughout this section we shall be using the same terminology as in Chapter III below 3. 5. Thus a G-cycle or G-manifold has singularities in codimension 1 at most, while a G-bordism is allowed to have singular ities up to codimension 2.
114
:heorem for cobordism without coefficients, Chapter II 2. 1, all the ;eometric constructions are carried out cellwise.
Therefore the proof
~eadily adapts to the present case and we deduce
!f.
Proposition 2. 1.
T' <J
T
then am: Uq(K', T') - Uq(K, T)
.s an isomorphism of abelian groups. Let ¢:
T
1
••
T
2
be a homomorphism of stacks over K. There is
In induced homomorphism
1Jl*:U*(K; T ) - U*(K, T) 1
2
defined blockwise
115
by relabelling; a
E
more precisely,
K, we relabel the block
(like in III).
If ~!(a)
if UK is a T1-cocycle [cobordism] and K, L) consists of a projection by means of cp(a):
~(a)
T
1
(a) .••
T
2
(a)
(a)
for each a
E
p~ : E~'"
=_
K, p~l(a) .,
nanifold, H<J);
is the resulting polyhedron, then the union
~'= __ ~'(a) is aT -cocycle [cobordism]. All compatibility conditions (b) for each a E K, ~(a) aEK 2 are ensured by !/J: T ••• T being a stack-homomorphism. Therefore he usual meaning of the notations); 1 2 we have the following: (c) P-l(L) = ¢.
IKI such that
is the interior
a
[a.
of a (p(a),
<
: a]p(a.
1
a)p~
1
1
q+dima)(with
(a.)
.,
1
fwo (p, q)-mock bundles Proposition assigns to each If> E
T E
2,2,
There is a functor n*(K, -) : S'" G.b* which
S the abelian group n*(K;
S the (graded) abelian-group
homomorphism
A linked stack of resolutions a covariant functor
p:
T IK
p is said to be p-canonical E
is represented
T
)q(-;
a E
K,
if p(a)
is a p-
K (in the sense of III §3).
is a stack of abe lian groups and p IK
resolutions such that, for each we say that
cp*,
p(a)
is a resolution of
T(a),
category of :ilK consisting of all stacks which are representable
=
1, 2, 3,
If
is constructed from
he theory of
T
1
<
q)-mock bundles in the usual fashion,
(p,
by
q)-mock bundles is that the former allows the coborthan the cocydes
while the latter is
Lemma 2. 4. There exists a coboundary homomorphism jq : nq(L; p) .•• nq-1(K, L; p) and a long exact sequence g f oq , ...•• nq(K, L; p) .•• nq(K; p) .•• nq(L; p) .•• vhere g is induced by (K, L)'" (K, ,0)and f is 'restriction
to L',
2
1
<
!!.
T E
a )* '
L full in K.
2
S', there is a spectral sequence,
n*(K-
T)
"
Suppose
Let J(t) be the t -nhd, of L in K and j its frontier;
i: J'" L the pseudo-radial
E(T),
retraction,
ir =
r and a (q - I)-mock bundle over (K, L), If
:(a); =}
It can be roughly described as 'pull
?TIt
If ~ is a (p, q)-mock
IUndleover L, form the pull back ir*(~), which is a q-mock bundle over Iver a, then
running E2 = HP(K' nq) p,q 'T
Definition of oq,
lack onto the boundary of a regular neighbourhood of L in K',
We aim to prove the following
~*(a)
~*(a)
comes from the block over
therefore it has a structure
is the block of 1i-*W
n-(j(t) n a), denoted
of p(ir(a»-manifold
and it is made
nto a p(a)-manifold by means of the stack homomorphism
(p(ir(a)< a), So the
'esulting object ir*(~)1K is a (p, q-l)-mock bundle which is empty over , The assignment [~] .•• [1T*WIK] gives a well defined morphism In order to prove the theorem we need some definitions and lemmlllq q() q-l qf : n L .•• n (K, L), If L is not full, one first subdivides baryLet p be a linked stack over K. A (p, q)-mock bundle ~ over ----:entrically once and then amalgamates.
Moreover the spectral sequence is natural on S', .-
116
-----.~--------
and
T
a ) "" T(a
Theorem 2. 3.
1
p, written
nock-bundle theory.
Proof. nq(a
(p,
lisms to have deeper singularities
The objects of S' will also be
= nq(point; T(a»
n;(a)
1
group of (K, L) with coefficients
then
is a stack of abelian groups, there is an induced graded {nq IK) and given by
T IK
o
Thus the main difference between the theory of (T, q)-cocycles
called nice stacks. stack on K, written
p),
~ , ~ /(K, L) are cobordant if there exists a o 1 . 17/(KXI, LX such that TJIK. = (_1)1~. i= 0,1.
he natural extension to the case of local coefficients of an ordinary
is a linked stack of
by p. We denote S' IK the full sub-
p-canonical stacks for each p
fhe (q-th)-cobordism
over a cell complex K consists of
canonical linked resolution for each a If
and to each morphism
e (see III §3 for the definition of e), A
~ ~
linked stack of resolutions
T)
p, q)-mockbundle
J,
117
Exactness is proved geometrically by mock-bundle arguments, which are all contained in Chapter II.
The only remark to make is the
following. Suppose K = a and ~ is a (p, q)-mock bundle defined on J = a ,a face of a. Then p gives morphisms going from the
a-
1
.asa block Ha) for each p-simplex i(a),
-q)-manifold,
1
if ~ is extended over a by the pull back construction,
ff~](a)
=
f
K
p-
l'
Therefore
~(a)
We associate to [~] a p-cochain
for each p-simplex
[~(a)]p
and ~(a) is a closed
p
~ is empty over K
etermines an element [~(a)] f nq(a). p p ~;] with coefficients n~ by setting
resolutions attached to the simplexes of J to the resolutions attached to a. Therefore,
because
a
a E
K.
p [~] ..• f[] defines the iso~. lOrphism h. Moreover the required convergence conditions hold and :is readily checked that the correspondence
the resulting block
~(a) has a natural structure of p(a)-manifold. q We remark that the coboundary homomorphism /) can be defined 00 aerefore E is the bigraded module associated to the filtration of also for the theory n* (K, L; T ) (T f S/K ) exactly in the same way as in I*(K;p) defined by the above proof. Therefore there is, for the theory n*(K, L; T), a long s sequence analogous to that of Lemma 2.4. However, although it is F n*(K; p) = Ker[n*(K; p) •.• n*(K _ ; p)]. p 1 immediately checked that the sequence has order two, there is no reason
to suppose that it is exact.
The argument which is used to prove Lemma 'he lemma follows.
2. 4 fails because in n*(K, L; T) two cobordisms having the same ends Proof of Theorem 2. 3.
cannot be glued together to give a cocycle. Given a linked stack of resolutions p /K there is an associated q f S defined like n above (T E S). We then have the
graded stack nq p
For each i
anonicallinked stack representing a given
=
1, 2, 3, let p. be an i-
T f
1
S'.
There is a commutative diagram of degree-zero
homomorphisms:
T
following: Lemma 2. 5.
!!.
a spectral sequence
E(p)
p is a linked stack of resolutions there exists
running
1
which t ..
1,]
Proof.
By Lemma 2.4, for each p = 0, 1, ...
we have the
g
. commutes with the coboundary operation and therefore there is an
f
a spectral sequence
118
)
1,2
lO
E(p)
t;,~ /;,: E(P3)
1
in which the chain complex {E ; d : p,q p,q q q fixed} is isomorphic to {CP(K; n ), /)} defined in Section 1. More p precisely an isomorphism h: E1 •.• CP(K; nq) is given as follows. 1 _ -p-q . p, q -p-q p . E - n (K , K l' pl. Let [~] f n (K , K l' pl. Then ~ p,q p pp pE(p)
homomorphisms
t* E(p
is the p-skeleton of K. The theory of exact couples yields 1
It is easy to see that
.,)
... n-P-q+l(K , K . p) •.• n-p-q+1(K . p) .•• n-p-q+1(K . p) /) P p-l' p' p-l' ..• n-P-q(K K . p) .•• p' p-l' ... p
map on cocycles.
lduced commutative diagram of spectral-sequence
exact 'p-sequence'
where K
is the 'relabelling'
In
the E2 -term
t:+'. is the homomorphism J t:+' .
1,
1, ]
HP(K; n~.) ..• 1
119
induced by the change of coefficients""
nq . relabel ---".-""
nq
whl'ch we know
E(G) -+ E(T) is an isomorphism
Pi Pj to lit! an isomorphism from the theory of coefficients in the constant case'
argument,
3
In order to prove the theorem we only need to show that
is an iso-
(]I
'
Corollary 2. 7.
If T E S' and we take the theory
n*(K) to be
H*(K, Z) (= simplicial cohomology with Z-coefficients),
then n*(K, T)
~~incides with the usual definition of simplicial cohomology with coefficients in T (see Section 1).
morfhism. 1.
sequences.
,J
r.*(K; T) -+ n*(K; P ) given by the relabelling map on the cocycles.
:
1
tt . il:lan iso-
morfhism for all (i, j) : 1 ::si <. j ::s3. Nowthere is a homomorphism (]I
n*T=n*(point, G).
Therefore the corollary follows from the 'mapping theorem b etween spectra
Then, by the usual spectral-sequence
2
on the E -term because
QI is an epimorphism.
There is
~lcommutative
diagram
Proof.
H*(K; Z) is the mock-bundle theory whose blocks are
oriented pseudomanifolds.
Since a G-pseudomanifold of dimension
greater than zero is bordant to ¢ for every group G, we see that the E1-term of the spectral sequence
E(T) reduces to the cochain complex
C*(K, T) considered in Section 1 and the spectral sequence collapses. where is
e
t
is epi, so
1,1
COBORDISM WITH CQEFFICIENTS IN A (PRE)-SHEAF
3.
(\I.
We are now ready to give a notion of cobordism with coefficients 2.
QI is a monomorphism.
the definitions that
~IK is also a
determines a class
[~]p
t 2,
Therefore, since
is also a relabelling map.
.,
But
Let [~] E Ker a.
It follows from
(p , q)-mO(~kbundle.
Therefore it
2
which is mapped to zero by t?,
is a monomorphisJ.. there exists a (p 2 ,q)-cobordism 1]
is also a (T, q)-cobordism.
Therefore
Because
3'
11 : ~ - (6.
(}' is mono.
. -
each a
E
.
If T/K IS a constant stack (1.e. -----
•.
K), then n*(K; T) co~ides
Proof.
u
T(a)
using an analogue of the Cech procedure.
polyhedron and F IX a presheaf of abelian groups.
= G for
•••
If K is a triangula-
tion of X, then, by the previous section, we have a graded group {nq(K, FK) }, where
FK is the induced stack on K.
spectral sequence E(G) running
,
: FK(a)
-+
FK,(a')
and we make ~"(a')
~'(a'), by applying this homomorphism,
into an FK,(a')-manifold,
i. e. f(a')
= F
of cohomolob'Ytheories
t : n*(K; G) -+ n*(K; T)
Ihe incidence numbers are preserved. in F-subdivision ['
of ~ over K'.
Two F-subdivisions
When
in the blocks, so that
The functoriality
of the presheaf
F ensures that the final object is an (FK" q)-mock bundle
~'IK',
called
~', 1;' of ~ over
are cobordant by the same construction applied to K x I and K' x I
modulo the ends.
120
,(~"(a')).
a,a 111 the blocks of ~" have been relabelled by means of the restriction homomorphisms, one takes care of the orientations
HP(K, n*(point, G)) '-'? n*(K; G).
given by relabelling.
Suppose K'
QlKK' : nq(K, F ) -+ nq(K', F ,) as follows: K K let ~qIK be an (F , q)-mock b~ndle. Subdivide ~ over K' and get K ~"IK' suchthat ~"(a') isanFK(a)-manifoldfor a'ca. Theinclusion t(' K') c t( K)' t . t· h h' sa, s a, gIves a res rlC lOn omomorp Ism
with i~*(K; G) _as defined il1JL?,. Faa'
Since n* (K; G) is a cohomolo~.,'y theory, there is a
There is a natural transformation
Let X be a
We define a homomorphism
The theorem follows . Corollary 2.6.
in a presheaf,
Therefore we have a well defined homomorphism
The induced homomorphism of spectral sequences
121
is a natural transformation
of cohomology theories,
inducing an iso-
morphism of the corresponding graded sheaves h
F'
. T* F
-+
S* F
The collection of gr oups and homomorphisms
Inq(K, FK1, O!K K'}'
indexed by the directed set of all triangulations
of X, is a dire~t system then h induces an isomorphism in local coefficients:
and we define the q-th (pl) cobordism group of. X with coefficients in F h*(X) : T*(X; F)
to be the gr aded gr oup:
-+
S*(X; F) .
4. QUASI-LINKED STACKS We now recall that a sheaf FIX open covering F(U)
=
(i = {U)
is locally constant if there is an
of X, such that, if U
E
(i and x
E
U then
We have given a definition of 'pl cobordism with coefficients in a sheaf' which works in the category of all sheaves,
is functorial on this
lim {F(V)} where V varies over the open neighbourhoods of x. category and extends the case of coefficients in an abelian group.
Un-
-+
It then follows that, if K is a sufficiently small triangulation of X, i. e. fortunately, we can be sure that the definition enjoys good properties the associated star-covering
is a refinement of
ct, the cohomology of K (e. g. spectral sequence) only when some special types of sheaves are
with coefficients in the stack FK coincides with the Cech-cohomology
involved.
Let us try to describe the source of this difficulty.
A mock
H*(X; FK) and the cobordism of K with coefficients in FK coincides
bundle and a stack, T, over a simplicial complex
with n* (X; F) by 2. 1. We call such an FKIK
in common: they are both 'local' objects in the sense that they are both
We say that a locally constant sheaf FIX
a limit stack for F IX.
is nice if it has a limit stack
which is nice in the sense of Section 2.
the category of pl
As in the case of stacks, to each sheaf F IX a graded sheaf
In~IX)
n~(U :) V) = F(U :) V)*; stant, then n~IX
defined by fl~(U)
functors defined on the category
=
there is associated
nq(point; F(U)),
u, V open sets of X. If FIX
is locally con-
A mock bundle takes its values in
manifolds and inclusions in the boundary, while a
stack of abelian groups ranges in the category of abelian groups and homomorphisms. T
coefficients,
is also locally constant.
1.
We have the following analogue of Theorem 2.3.
~.
K, have a main feature
Thus in order to have a notion of mock bundle with
which has good properties,
a notion of r(a)-manifold
(a
what is n~eded is the following: E
K) for which the corresponding
mock bundle theory is 'cobordism with T(a)-coefficients'.
2. a recipe for associating a r(a I-manifold to a r(a I-manifold Theorem 3. 1. On the ca!e~()ry of nice sheaves there is a natural in a way which is functorial on K. 2 1 spectral sequence running h . ~--_ ..-~~"". -~---~~ T erefore we see that the difficultIes ansmg from our defmition can be traced to the lack of a bordism theory of G-cycles which is functorial (with respect to G) on the cycles, rather than only on the bordism The proof is the same as that of 2. 3, using limit stacks.
classes.
A direct consequence of the 'mapping theorem between spectral
be fulfilled and since some of the cases look rather
sequences' is the following comparison theorem Proposition
122
3. 2.
Let F*IX
be a nice sheaf.
Now in many instances it happens that conditions 1 and 2 can interesting,
we will
make a detailed discussion of them. Thus, in the remainder of the chapter, we abandon the point of view If h: T*(-)-+S*(-)of setting up a theory of local coefficients in generality. Instead we dis-
123
method and give examples to show how local coefficients may at times
and nq(a < a ) : [M] .•• [S(a < a )M] • s 1 2 1 2 As an illustration of the general setting described above, we dis-
reveal relationships between local and global properties of a space.
cuss some examples associated with Poincare duality.
cuss, along the lines of the above remarks,
continue to work only with pl
some generalisations
of our We
cobordism, but everything that we say
remains true for an arbitrary
geometric cohomology theory.
Let p, p' be linked resolutions and f : p ••• p'
Let (P SP) be a Z -manifold of dimension
,
(a)
For each b
E
P
a chain map. f
B , P
=
0, 1, 2, 3 either
P
f(b )
=
If n
0 or
one of ±f(b ) is in B'P. (b) L(~,
P
For each link L(b , p) let fL(b , p) be obtained from
p) by the fOllowing process.
j
£'(K) of
local m-homology on K.
P
P
=
triangulated by (K, SK)(SKC K) and consider the stack £
is said to be quasi-linked if the following conditions are satisfied: P
m (see III 1. 1(2)),
n
£.(a'
<
P
Let V C L(b , p) be a stratum
=
2, then a presentati?n
=
of £. is the following:
(a)
£.(a)
free abelian group, F(X~, on one element
(b)
if a', a EK _ SK or a', a ESK and a'
a) is the isomorphism mapping xa'
(c)
<
if a' ESK, a E K _ SK, a'
j
<
xa
a, then
to xa;
a, then £.(a'
<
a) is one of the
labelled by b : if f(b ) = 0, remove V from L(~, p); if isomorphisms x .••±x , depending on which 'sheet' a belongs to. We j' j j j a' a f(b) EB'l, relabel V by f(b); if -f(b) E B', reverse the orientation call 'positive' (resp 'negative'Hhe sheet corresponding to the ,+' sign .. p p --of V and then label -V by _f(bJ). Then we requIre fL(b , p) = <5L(<5fb ,f:. (resp. ,_, sign). for one of <5= ± . For a Z -manifold, £. can be presented as follows. If a E K- SK, 3 a a To each (p, n)-manifold M there is associated a (p', n)-manifold then £.(a) = F(X ); if a ESK, then £.(a) = F(X , x/ The morphisms a 1 1 f(M) constructed on strata in the same way as fL(b , pl. There is a are so defined. category Q, whose objects are short linked resolutions and whose morphisms are quasi-linked maps.
A quasi-linked stack of resolutions
over a ball complex K is a covariant functor S = SK : ~ .•• Q.
(a) (b)
if a' ESK, and a EK _ SK; a'
of the following isomorphisms,
p~:E~"'IKI such that, for each aEK, p~l(a) istheinteriorofa~l (S(a), q + dim a)-manifold WJJ, with oW1J = ~ [ai:a]S(ai < alp ~ (a/
{xa' (1)
.••x
x2
•••
<
(2)
.•• _x
K, L (L
C
(b)
K) with coeffic ients in S, depending only on S, a spectral sequence running
IK I, IL I.
a'
xa
x2
•••
<
a) is one
a belongs to:
a'
0
x2
=
-xa
•••
1, 2, 3) will be referred
to as 'sheet (i)'. Associated to £. there is the following quasi-linked stack, A, on ,
the Z -manifold 3
,
We write
id
K: if a EK - Sl.{Athen A(a)
a E SK then A(a)
nq(p,.£)
is F(x
..•
aala l'
x)
2
aa
>-40>
is F(xT)
F(x , x ); A(a'
= lim nq(K, A).
1
2
<
a)
-.
=
F(xa); £.(a'
<
if ) a.
Here .£ stands for 'local-homology
sheaf'. where oq is the stack of abelian groups over K defined by s
124
a)
(3)
a
1
The sheet for which the assignment (i) holds (i
an abelian group nq(K, L; S), the q-th cobordism group of
<
{x~''''O
in the same way as the theory of cobordism with coefficients in a linked (a)
a, then £.(a'
depending on which sheet {xa,
a
1
a'
I
The cobordism theory of (S, q)-mock bundles is developed exactly stack of resolutions (see §2). In particular we have
a then £.(a'
is the canonical isomorphism;
If K is
oriented, an (S, q)-mock bundle ~q over K consists of a projection
<
if a', a EK _ SK or a', a EK and a'
With the above notations we have the following
125
Proposition 4. 1.
n == 2. In this case SP is an orientation-type singularity.
Proof. Let [~]
E
There is a duality isomorphism:
EW is a manifold (see Chapter II).
nq(K, A). The total space
Moreover E(~) is oriented, because it is oriented over both K - SK and SK and the action of
£,
makes the orientations of the blocks over the
positive sheet compatible with those of the blocks over the negative sheet. Therefore we regard 'glueing map'
1/1
E(~) as an oriented bordism class and define a
like in Chapter II.
The proof that
1/1
is an isomorphism
reduces essentially to the proof of Poincare duality given in II and we omit
it. n 2:: 3. Again, for the sake of simplicity, we only discuss the case n == 3. The general case is dealt with using the same arguments. Let [~] c nq(K, A). The block of ~ over a simplex a' E SK is a pl a a manifold, each component of which is labelled by either x ' or x ' and I
2
at most two non-empty blocks merge into it, so that no singularities created in the glueing process (see Fig. 18). case, the signs in the stack-homomorphisms
are
Moreover, as in the above Fig. 18
ensure that the orientations
of the blocks are compatible in passing from one sheet to another across SK. Hence the total space E(~) is an oriented pl
manifold and the
operation of 'glueing up and disregarding the labels' gives the required homomorphism 1/1. We now prove that
1/1
is an epimorphism.
Let f : Wm+'4K
a simplicial map representing an oriented bordism class of P. Cohen's generalized transversality
be
Then
theorem (see IV 1. 1(2)), together with
(b)
(a)
fir 1(L -
bundle, where L
Int L ) is the projection of a pl oriented q-mock a is a subcomplex of L triangulating a product neigh-
a bourhood of SP in P, i. e.
I L a I = SP x
cone (3 pts).
In order to avoid
technical details, we assume that L is the product cone-complex . a SL x cone (3 pts) and write L1 for L n Sheet (i) (i == 1, 2, 3). a a
126
is an oriented manifold with bOWldary rl(aL ) a a ) is the projection of a q-mock bundle over the cone com-
and T Irl(L o plex L = {a x cone (3 pts) : a E SL l. a . Now we take boundary collars rl(aLJ) x Ie rl(L ) (j = 2, 3) a a and 'stretch' them along the cone-lines of L until f is replaced by a m+q a map f' : W ...•1 L I such that, with obvious meaning of the notations: (a)
the subdivision theorem, tells us that there exists a cell decomposition (L, SL) of (P, SP) and f"" f, such that
rl(L)
(b)
f' "" T mod L - Int L . a . f,-I(SL) = rl(aLJ) x 1 c r1(aLJ x I) (j a a
=
2, 3). (See
Fig. 19.) Then, for a' E SL and f,-l(a') "* li!, we label the block f,-l(a') a' by xl if the two non-empty blocks merging into it lie over the sheets (1) and (2) respectively;
a
E
otherwise we label f,-l(a')
L - SL we give f,-l(a)
the label x.
a
by xa'. 2
For each
In this way f' : Wm+q ...• IL I
gives rise to a q-mock bundle associated to A , L
This shows that
lJ;
127
(*) 0 (SM) + II(SM, M) = 0 (M)ISM, where 0(-)
is the orientation cover of (-), II(SM, M) is the normal
cover of SM in M and + denotes sum of isomorphism classes of Z 2
bundles. if f is null-homotopic, we can take SM = 50 and
In particular,
M becomes an oriented manifold. Now Proposition 4. 1 applies. 5. A FINAL EXTENSION
AND EXAMPLES
SL Let
a
be the category defined as follows.
linked resolutions;
a map of resolutions
morphism of Ci if f quasi-linked map.
Fig. 19
f: p ••• p'
The objects of Ci are (p, p' E Ci) is a
=
nf , where n is a positive integer and f is a o 0 Clearly f and n are uniquely determined by f.
o
Let f be as above, then to each (p, n)-manifold
is an epimorphism.
The injectivity of
l/I
follows from the same argu-
associated a (P', n)-manifold f(M) defined by f(M)
=
M there is
disjoint union of
n copies of f (M). o An Ci-system on a ball complex K is a covariant functor
ments applied to P x I. The proof of the proposition is complete.
Another example of a polyhedron, whose local homology gives rise S = SK : ~ -+ Ci. Let ~a be the class of S(a)-manifolds and :m: = ~K) = ua~. a If K is oriented then an (S, q)-mock to a quasi-linked stack in a natural way, is provided by any unoriented ---'----bundle ~q over K consists of a function ~: K -+~, such that, for each a E K, ~(a) is pl manifold Mm. We leave the reader to make the obvious definition of nq(M, £) in this case, while we establish the following fact about M, an (S(a), q+dim a)-manifold, labelled by a, with which is an easy consequence of Proposition 4. 1. a~(a)
Corollary 4. 2.
Suppose that the orientation cover
is iS0I!l0rphic to f*(1)), where the circle
m
0 (M ) of M
1) is the non-trivial double covering of
S1 and f: M .••S1 is a map.
U
T<
[a:
T]S(T,
aH(T)
a
where the obvious identifications are made in the union. We note that an (S, q)-mock bundle does not have a total space or
Then there is a duality iso-
a projection.
morphism,
=
But nevertheless we can set up a theory of (S, q)-mock
bundles in analogy to the usual case: restriction,
The various notions of isomorphism,
cobordism etc. for (S, q)-mock bundles are obtained by
obvious modification from the definitions for ordinary mock bundles. Proof.
We regard
M as a Z -manifold in the following way. 2
1
leave the reader to write down the details and to establish the existence
-1
of: to a point XES, then take SM = f (x) o 0 and give orientations to M - SM and SM. The fact that SM is orientable First we make f transverse
follows immediately from the equation.
128
We
(i)
an abelian group nq(K, L; S), the q-th cobordism group of
(K, L) with coefficients in S, which depends only on S and
ILie IK I. 129
(ii)
a spectral sequence
VII·The geometry of CW complexes
The sheaf of local homology, considered in §4 for a Z -manifold, n
provides examples of
Here
we mention three: In this final chapter we draw together all the ideas of the previous (a) Let
X = homology n-manifold, i. e. H.(X, X-x; Z) = H.(Sn; Z). 1
1
Pz be any short linked resolution of Z. Then, for each a
/KI =X, we set S(a)=PZ
and SPz(a<
T)=mo
id:pZ-pZ
E
K,
where
m : Z - Z is the multiplication given by the local homology stack.
Pz
) is independent of the presentation
. theory in an essentially unique way. coefficients,
cohomology theory is a geometric Thus the geometric definitions of
operations etc. all apply to an arbitrary
theory.
Using achieved by examining the geometry of CW complexes.
the spectral sequence (ii) as in the proof of Theorem 2. 3, we are able to conclude that nq(K; S
chapters by showing that an arbitrary
Pz and
therefore provides a good definition of cobordism with coefficients in the
a new concept, that of a transverse geometric properties transversality
sheaf of local n-homology of X.
We will define
CW complex, which has all the
of ordinary cell complexes.
dual complex and transversality
This is
In particular,
constructions can be applied.
it has a The
theorem (in §2) is a version for a CW complex of the
theorem in part II §4. However the proof uses even less and is ele(b)
X = rational homology manifold.
1. e. H. (X, X-x; Q) = 1
mentary!
H.(Sn; Q). In this case fix a linked resolution 1
nq(K; Sp ) as in (a). Again nq(K; S
Q
P for Q and define Q ) does not depend on the par-
PQ
plex, then the subcomplex
X is a polyhedron with only two intrinsic strata.
Again
there is a good definition of cobordism with coefficients in the sheaf of local homology. We leave details to the reader. Finally, we conjecture that, at least when there is a 'good' definition of cobordism with coefficients in the sheaf of local homology, there is a 'Zeeman spectral sequence', cobordism with local coefficients.
cf. [1], relating bordism with
f : M - X can be made transverse
A map
to X (X) (in fact the transversality
theorem does exactly that) and the transverse
map is determined by its
values near f-I X (X). In this way, an arbitrary
spectrum is seen to be
a 'Thom' spectrum for a suitable theory (with singularities),
see §§4
and 5. Another consequence of this chapter is that CW complexes, already useful as homotopy objects, now have a beautiful intrinsic geometric structure.
REFERENCES FOR CHAPTER VI Dihomology III. Proc. Cam. Phil. Soc., (3)
13 (1963), 155-83.
This has strong connections with stratified sets and
the later work of Thorn, see §3. We intend to write a paper [5] examining transverse
E. C. Zeeman.
(X) c X*, consisting of dual objects other
base of a Thom complex behaves with the whole complex.
Proposition 4. 1 gives exactly this
for a Z -manifold. n
[1]
X
CW complex and X* its dual com-
than the object dual to the basepoint, behaves with X exactly like the
ticular resolution. (c)
If X is a based transverse
CW complexes in greater detail and showing that they have
all the properties enjoyed by cell complexes and semisimplicial plexes.
com-
In particular block bundles or mock bundles with base a trans-
verse CW complex can be defined and have good geometric properties (see also §4 of this chapter).
130
131
The main theorem in §6 is that the stable homotopy category is
C*(b.)=cl{b. 1
1
o
equivalent to the category of geometric theories (theories of 'manifolds' with singularities,
labellings etc.,
formally inverted. of singularities,
see Chapter IV) with 'resolutions'
b.1 lb.1 =
b.1
n
1
st(b.,
of theories is equivalent
III
sY)
=
v
cl
to a relabelling followed by a 'resolution',
see the examples in §6. Thus a theory has products if and only if the product of two cycles in the theory rhen if we write
can be relabelled and then 'resolved' to give a cycle in the theory, see
v
v
1
Throughout the chapter we will use the standard notation for CW complexes.
{b. b. o 1
1
lb. = b.
b.
111.
n
v
C(b.) = b.C(b.),
also the final remark of §6. i
1
n addition to the usual
Thus any theory has cycles unique up to resolution
and any natural transformation
b.}
0
v.
v
st(h.) = b.lk(h.),
C*(b.) = b.C*(b.),
1
1
1
some j} J
1
1
1
1
C(i;i) = C(bi) - C(bi), etc.,
j
Thus X, Y etc. denote CW complexes, e , e , e , e etc. h' . I homeomorp h'Ism 1 2 ere IS a canomca denote cells; all cells have given characteristic maps denoted h : D ...•X, h : Dj -+ X etc., where Dj = [- 1, l]j C Rj. We denote 1
1
h (0) 1
€
2
e
1
C
a simplex.
e1
X by
in analogy with the notation for the barycentre of A plan for Y is a map d: Y -+ sY such that d
Other notation is in §l.
--
hen {Y, {b.}, d)
We would like to acknowledge a helpful conversation with
1
"'"
1
1
{b.} and plan d. We
---
1
dY (the derived of Y) = {d-1(a), (] € sy)
BIDLDINGS
lndthe dual building
In this section we describe a general structure which allows a dual structure to be defined
The examples will be used in later sections.
Let Y be a space and {b.} a partition of Y subsets.
-
0
(C(b.)) = b. and
hen have a new partition
C. T. C. Wall at the beginning of the work of this chapter. 1.
is a building with bricks
-1
Write b.
1
*j
< b. if i 1
1
and b.
C
1
-1
y* = {Y, {b:"} , d } where b:" = d 1
into disjoint
b.. Define the simplicial
,..
Notation.
1
1
We write b. for d 1
0
.•...
(C*(b.)). 1
-1 "'"
(b.). 1
complex sY to have for typical n-simplex a string Examples 1.1.
<
(b.
1
o
b.
1 1
< ... <
b. )
n
(b.) of sY and
b. b. '" b.
1
In
10 \
1\ for the
Write
for the simplex (b. 10
< b. < ... 11
b. ). In
2.
I) then
1
{b.1 b.1 '" o
1
b.1
n
lb.
1
n
= b.1 }
Our notation is then
Suppose Y is a CW complex in which the closure of each
:ell is a subcomplex.
cl
Let
{Y, {e.}, d) 1
:onical extension.
Then
{e.} be the open cells of Y (which partition 1
is a building, where d is defined by inductive
e.I
is the centre of e. and the notation is con1
iistent.
and
132
dK
{K, {(1 ), id. } is a building with
Iricks the open simplexes and plan the identity.
sY has the structure of a cone complex in the sense of McCrory [3];
=
=
:onsistent with the usual notation for barycentres and first deriveds.
in particular we can define C(h.)
Let K be a simplicial complex and sK
he usual first derived of K. Then
1
and faces given by omitting members of the string. vertex
1.
133
3.
Supposep
:E
K is a mock bundle, over a simplicial
-+
llso transverse.
complex or a cone complex, in which each block has a collar and the
We say that the CW complex X is transverse,
or that X is a
projection is defined by mapping collar lines radially (the 'canonical
rcw complex, if each attaching map is transverse to the skeleton to
projection').
uhich it is mapped.
sE
C
Then
E then
b.
1
1. e.
Note that if b. is an open block of
Transversality
1
= cl (b. - collar).
Suppose X is a stratified polyhedron in the sense of Stone X is partitioned by open manifolds, the strata,
and provided
I aM
the strata for bricks in which the plan is obtained by using the mapping of a regular neighbourhood.
b. =
Then
Suppose X is a TCW complex manifold.
map.
Corollary 2.2.
Any CW complex gives rise to a TCW complex
If the same homotopy type obtained by inductively homotoping attaching
1
the closed stratum.
Proof of the transversality
theorem.
Since im(f) is contained
n a finite subcomplex of X, we can assume
2. TRANSVERSALITY
X is finite and proceed by
nduction on the number of cells of X. Suppose X Transversality smooth or pi
for CW complexes works nicely with either the
categories.
category but a
Choose a collar
c for
Let M be a closed pl
manifold and X a CW complex.
if for each cell e.
-----
= Xo u er
aM in M and by a preliminary
In this section we choose to work with the pl 'el aM assume that f is constant on collar lines.
similar treatment is possible with the smooth category. ;tandard transversality
f : M -+ X is transverse
1
E
X either
f
-1
Suppose
Then there is a homotopy of f reI aM to a
cl (b. - neigh- naps to make them transverse.
1
bourhood of previous strata),
is transverse.
ransverse
with a system of regular neighbourhoods. There is also a local triviality condition which does not concern us. Then X defines a building with cylinder structure
theorem 2. 1.
md f: M -+ X a map, where M is a compact pl
1
4.
[61.
IE, {open blocks}, p} is a building where
dK (the usual first derived).
= 1
(e.)
A maPransverse ¢ or
liagram
to
e.
homotopy
Now apply the
theorem (see Remark 2. 3 below) to make f
in e., by a homotopy of f reI im(e).
1
1
t"
D.(E)
We now have a
there is a commuting diagram T cl(T.)
t
l~
~ .
/.
1/
fl ~
D. 1
hi
f~/h:1 X
X
vhere D. (E) is a small disc in D. centred on 0 and t is a trivial bundle. 1
1
= f1
where T.
jection of a pl
(e.), 1
h. is the characteristic
map for e., t. is the pro-
1
1
bundle (necessarily trivial) and cl(T.) All
zero in M. Notice that this implies that T.
1
=
1
M = M U T. o 1 attached to M
1
t~I(O) is a submanifold 1
i.
by T. x an .. o 1 1 In the case that aM"* ¢, we insist that cl (T.) meets A
A
134
A
C
aT.1 has codimension
1
to the whole of e ..
in itself
1
1
1
IY using the collar in the usual way, to a homotopy of f which keeps handle'lM fixed, we have the same diagram with D. replacing A
A
Now write aM in a
I
Hence, by
1
e.
obtained by expanding D.(E) onto D. and using h.) and then extending,
All
subtube T' x D.1 where T'
1
is already transverse
has codimension :omposing flcl(M - im(c)) with a standard homotopy of
Notice also that if X = X o u e.1 then x D., where T. x D. has the form of a 'generalised
of M of codimension
1I0reover flim(c)
O. Thus f aM is
II 1
=
liT = T x aD. (T 1
cl (aM - T). Then M
=
-1
t
D.(E). 1
1
(0) as usual), M
0
=
cl (M - T),
is a compact manifold with boundary 0
~1 U liT• Now from definitions and the diagram both f
1M
1
and f I liT
135
are transverse to X
(and to X). Thus f 13M is transverse to X o 0 0 and by induction we may homotope f further reI 3M to make it transo verse, as required. Remark 2.3.
We only used the simplest
pl
Corollary 2. 6. to a transverse
map.
The homotopy can be chosen to keep fixed a sub-
complex on which the map is already transverse. Proof.
transversality
Any map between TCW complexes is homotopic
Use the transversality
theorem inductively to shift cells
The lemma and induction ensure that a cell theorem, which has an elementary proof using the fact that the pre image to make the map transverse. is already transverse on its boundary. of the barycentre of a top dimensional simplex by a simplicial map is The proof of Lemma 2. 5 uses the description of M as a framified framed - a fact that has been known for about forty years! The corresset which is given in the next section and will thus be left for convenience ponding smooth theorem is also elementary, using Sard's theorem. Now let M+
=
M
OMx I, i. e. M with a collar glued on
U
until the end of that section.
3Mx 0
'on the outside', and let q : M+
-+
M be the map which projects the
collar back onto 3M. If f : M -+ X to be f
q. We say f : M -+ X is weakly transverse
0
--~-----
if f
+
is trans-
verse. Thus f 13M is transverse but some of the tubes T. might only 1 have codimension 0 in OM. An example of a weakly transverse map is a characteristic
3. FRAMIFIED SETS
is a map, then define f+: M+ ..• X
map for a cell in a TCW complex.
Roughly speaking, a framified set is a stratified set in the sense of Stone [6] in which all the block bundles are trivialised.
The precise
definition is similar to the definition of killing in Chapter IV, and in fact
In fact it can easily a precise connection will be formulated in §4. The definition is by in-
be seen that X is a TCW complex if and only if all the characteristic
duction on the length of filtration.
maps are weakly transverse. A map f : X f
0
h. : D. 1
1
-+
-+
Y between TCW complexes is transverse
Y is weakly transverse
for each characteristic
if
map h. of
Definition 3. 1.
A framified set ~ of length n consists of
(1) A polyhedron X with a filtration
1
X.
{X = X :::>X :::>... 1
'rheorem 2. 4.
2
X
:::>X +1
n
n
=
¢}
There is a category TCW consisting of TCW --------~-~ complexes _
suc h th at Xi - X is a manifold for i = 1, 2, ... , n. H1 (2) A regular neighbourhood system for the filtration,
~E_~q~'yalence of homotopy categories.
1 :S j :S i :S n, (see Remark 3. 2 below).
TCW is closed under cross
N.. , 1, J
prodl.l~t:.L~<:dg_e product, factorisation of a subcomplex and smash product
(3) For each i:S n a framified set L. of length i - l.
!EE-a!~1:!:~~_!!.j.~c~ose~nder
(4) For each i:S n an isomorphism of filtered sets
-1
Proof. definitions.
suspension (smash product with SI).
That TCW is a category follows from 2.5 below and
That TCW C CW induces an equivalence of homotopy
h. : {N. 1:::>N. 2 :::>... 1
1,
=
1,
1, 1
- 1
trivial verification.
the cone point added as the final stage of the filtration.
1,
1):::>pt.}
x C.
where C
'
C(L ..
1, 1
categories follows from 2.2 and 2.6 below. The rest is a matter of
- 1
{C(l. 1):::>'"
N.. } ~ N..
1,1-
is the cone on L. with -1
The cone flag C. is a fibre or model for the framified set X Lemma 2. 5.
f: M -+ X, g : X -+ Y are both transverse
where M is a compact pl Then g
136
0
manifold and X, Yare
maps
TCW complexes.
-1
--
---
-'
and the framified set L. is a link for X. There are obvious notions of -1
restriction
--
-
of a framified set to a suitable subpolyhedron and of product
f : M -+ Y is transverse.
137
of a framified set with a manifold.
A pl
homeomorphism is an iso-
Moreover, by uniqueness we can choose the structure
T.
on aT to be the
morphism of framified sets if it commutes with all the extra structure.
product of the framification of S. 1 with
In particular two isomorphic framified sets have the same (or identi-
agree and extend the strata to M by adjoining the 'cones' on their
fiable) system of links.
intersection with IlT. L e.
The final condition is:
(5) h. restricts
to an isomorphism of framified sets
1
{N.1,1:J N.1,2 :J Remark 3.2.
...
1
N..1,1-I}
3!
N ..
x
1,1-1
Choose the indeXing to
M. == (M ). U T x (C(S. 1)' - cone pt. ).
a
J-
1
1
L ..
Finally the isomorphism
A regular neighbourhood system is constructed
inductively by defining N == X and N . is a simultaneous system n,n n n,J of second derived neighbourhoods of X in X .. Then define n J X: == X - int(N .) and proceed with the construction for 1
J-
ht is provided by the chosen product structure
on T. Induction now gives all the structure of a framified set to M. Uniqueness is clear.
n,1
X'1 :J X'2 :J •••
:J X'n-l :J ¢.
Existence and uniqueness of regular neighbourhood systems thus follows from the usual regular neighbourhood theorem. Notice that a framified set is a building with bricks the strata {X. - X. I} 1
and plan defined by using the cone structures
1-
(see §l
Example 4). We are not interested in the specific ordering of the strata of ~ but only in the partial ordering given by the geometric structure of X, and we will allow an isomorphism of framified sets to change the order.
The importance of framified sets lies in the following theorem,
which is essentially an observation. Theorem 3. 3.
Let f: M ...•X be a transverse
map to a TCW, Fig. 20
then M has the, structure of a framified set determined up to isomorphism by the map f. Proof.
We can also describe this framification (at least as a building) Since the image of f is contained in a finite subcomplex
quickly as follows.
of X we can assume without loss that X is finite and proceed by induction on the number of cells of X. Mt == f-1(e) and Nt 1 == cl(f-1(e»
,
X -- X*1 :J X*2 ::>•••
In fact we will produce a framifica-
tion of the same length as the number of cells. (i. e. Mt ==
Let X == X
a
u e.
Define
of and Nt, 1 == T in the
Now let M == cl (M - T) then M a a x S. 1 have the structure of framified sets by induction.
138
T
J-
::>X* t-- e
A
be the corresponding filtration of X.
Then Mt == f
-1
(X;).
In other words
the building is the pull-back by f of the dual building to X.
notation of the previous section). and IlT 3!
Take the dual complex to X and let
Notice that the 'models'
C. are just the closures of the cone
-1
flags
139
xn .
e. n (X'" ~ X'" ::J ••• 1
1
2
But all the product structures
1
required product structure
are coherent in the Dk factor and the
on T. is seen. 1
Since the models depend only on X, we say that the framification of M is modelled_on X. It is clear that there is a 1 - 1 correspondence
4. MANIFOLDSAND MOCK BUNDLESMODELLED ON X
between transverse maps f: M - X and framifications of M modelled on X.
We now rephrase the results of §3 by omitting the first stratum
This observation will be developed in the next section when we
'classify' homotopy classes of maps from one TCW to another.
To end
throughout.
suspension and thus carries
this section we will prove the lemma left at the end of the last section.
1,
l
we can fwd an isomorphic system with smaller closed strata
over to stable maps.
as the first cell in X) and let
and larger cone flags (d. Stone on 'minidivision' [6]); this has the effect,
of the duals to cells other than
for the framification given by a transverse
associated stratification
•
map f; M ...•X, of replacing
A.
+
A
f
the nelghbourhoods T. x D. by nelghbourhoods of the form T~ x D. A
1
A
1
1
where T: ~ T. and D:t-= D. u collar, 1
1
1
1
1
and it is not hard to see that we
A
o
-1
M:)
Let e. be a typical cell of X.
(X (X)).
and write
E
1
= cl(x(X) 1
C.
X be the basepoint (regarded
x (X) be the subcomplex of X* *. If f : M ...•X is transverse
of M starts
X(M, f) where
1
.,
1
1
Using the fact that L.
1
of a framed X manifold.
.
a framed Xo manifold.
We call this framification an extension of the original one.
11
=
1
1
1
aD 1.. The set of basic links {L.) 1 C
(see IV §3) called free
S., we have an intuitive notion 1
The precise formulation is in terms of
killing as in IV §4. Suppose X is finite and X inductively that framed
X(M, f)
Collapse the notation of §3
defines a theory of manifolds with singularities
X
then the
and we label the cone point bye .. L. is in
fact a framified set embedded in SI'= X manifolds.
consisting
n e.), then C. is the cone e.L., where L, is
the link associated to e
can choose all the collars so that there are diagrams
*
Let X be a based TCW. Let
First observe that, by inductively choosing collars on the frontiers of the N..
The idea is to obtain a formulation which is invariant under
= Xo
U
e.. 1
Suppose
Xo manifolds have been defined so that L
is i The theory of framed X manifolds is the theory
obtained from this theory by killing L. and labelling the new stratum of Proof of Lemma 2. 5.
Let e
be a cell in Y and Tk = (g f) Choose an extended
This means that we can regard
made of generalised handles of the form T ~x D:. 1
is transverse
+ n Tk i
D
aDt
in a similar subtube.
as included in M as pt. x D:
1
6:1 x D:'-) n T. = (Q 1 1
C
T~ x D:. 1
x T~) x Dk 1
= Q' x Dk say.
140
1
(from the definition of transversality
M as
Since go f I : D: -Y 1
1
Here we are regarding Thus
1
singularities framed
bye .. In general define a framed 1
X' manifold where X'
C
X manifold to be a
X is a finite subcomplex.
From the
definitions of killing and framified sets, it is easy to see that a framification of M modelled on X is equivalent to a framed
for TCW's) we have
= Q x Dk say where Q is a manifold with boundary and
Q x Dk meets
ek.
k is a product Tk x Dk of codimension zero
We have to show that cl(Tk) in M which meets aM in a similar subproduct. framification for f: M -. X.
-1
0
X manifold
embedded in M. From Theorem 3. 3 and the transversality
theorem we
have:
Dt
Proposition 4. 1.
There is a 1 - 1 correspondence
between the
set of homotopy classes of maps [M, X] and the set of cobordism classes of framed X manifolds embedded in M. In particular n
UX means the group of cobordism classes of framed
1T
(X) ~ Un where n X--
X manifolds em-
bedded in Sn.
141
We can extend the proposition to maps [Y, X] where Y is an unbased TCW using an extension of mock bundles to TCW's. be a TCW then a subset
E
C
Let Y
enlarged by adding a trivial I-disc bundle.
We thus have the stable
version of 4. 1 and 4. 3 (for details of the stable category see Adams [1]):
Y is the total space of an embedded mock
bundle (of dimension -q) provided that for each cell e
i
E
Proposition 4. 4.
Y there is a
diagram
There is a 1 - 1 correspondence
between stable
homotopy classes of (based) maps Sn -+ X and cobordism classes of stably framed
X manifolds.
There is a 1 - 1 correspondence
stable homotopy classes of stable maps
{Y, X)
and framed
between
X mock
00
bundles over Y - * (i. e. embedded in S Y - *). Remarks.
1.
direct generalisation
= 1
where M.
The above notion of a mock bundle over Y is a
of the definition for cell complexes in Chapter
n.
h~1(E) and is a proper submanifold of D. of codimension 1
1
2.
q. The following proposition is a generalisation of Lemma 1. 2 of Chapter II and is proved by a similar argument to Lemma 2. 5. We omit the
We have been deliberately careless
about dimensions; 3.
details.
this will be remedied in the next section.
See Example 2 at the end of the next section for a clarifica-
tion of the relation between this representation Proposition 4. 2.
Let E
C
and 'killing'.
Y be an embedded mock bundle and
f : M -+ Y '!-.i!'ansy~!,~emap, then f-1(E) is a proper submanifold of M
5. THE CYCLES OF A HOMOLOGYTHEORY
<2!... codi}ne.!lsion q. The proposition implies that mock bundles can be pulled back and hence give a contravariant functor on TCW. There is an obvious extension of the notion of mock bundle to mock bundle with singularities
(or rather uninformative)
and framed mock bundle. The next proposition
At the end of the last section we had represented bundle theory.
In this section we extend this result to arbitrary
spectra
and observe that the corresponding homology theory is the corresponding bordism theory.
is, like 4. 1, essentially an observation:
a cohomology
theory, which came from the suspension of a CW complex, as a mock
The results are most elegant for connected spectra
when we will be able to represent Proposition 4. 3.
There is a 1 - 1 correspondence between
trans~-.!'~e maps Y - X and framed Hence [Y, X] ~ nX(Y)' where
X mock bundles embedded in Y.
h () by n-manifolds with singularities. n For non-connected spectra we will need manifolds of non-constant dimen-
sion.
Uniqueness of representatives A spectrum
frameQ X mock bundles in Y.
~----
There is a based version of 4. 3; [Y, X]* = QX(Y) where 0X(Y) denotes cobordism classes of framed
X mock bundles in Y - * .
We will now stabilise these results.
Let SX denote the (reduced)
will be discussed in §6.
We follow Adams' treatment of the stable category, [1].
0X(Y) denotes cobordism classes of
X is a sequence
ix.,1 q.,1 i ~
0)
of based CW com-
plexes and based maps where q. : SX. -+ X'+
is an isomorphism onto a 1 1 1 1 It is connected if X. has no cells other than * in dimen-
subcomplex. sions
<
----
1
i.
= * u ei1 u e 2 u •..
suspension of X which is again a TCW then X(SX)= X(X) and if f : Sn -+ X is transverse then so is Sf: Sn+ 1 -+ SX. Moreover
Make X. into a TCW and, by further homotopies, ensure that each
X(Sn+1, Sf) = X(Sn, f) and the only difference between the framification
cell e ,e
of Sn+1 given by Sf and that of Sn by f is that all the bundles are
homotopic ally !). Then X(X.) =
142
Suppose X is connected and, for some i> -
1
2
0, X.
1
.
3
...
wraps non-trivially around e1 (geometrically, 1
e 1 u ...
1
not
will have constant c odimension
143
i in X.. Now SX. is again transverse 1
1
and since SX. C X'+l 1
1
we can
make X + 1 transverse keeping SX fixed and, by further homotopies, i i ensure that each cell in X. - SX. wraps non-trivially around Se . 1+1
1
1
Thus X(X. 1) again has constant codimension (this time i + 1). Pro1+ ceeding in this way we can make each X. transverse, for j 2:: i, with l
SX. C X'+l J
J
and X(X.) of constant codimension
Now define the geometric homology theory associated to ~ by
taking for basic links all the links defined by X. for j l
2::
SXj C X + l' Each link i is stably framed and it makes sense to talk of stably framed ~ manifolds and from the results of §4, Chapters II and IV and definitions we have:
by the spectrum
The homology and cohomology theories defined ~ manifolds.
In other words ~ theory is framed bordism with M 'killed', see Chapter IV §4. This example is the germ of the whole construction. which, when killed, gives the new theory.
in detail.
Moreover by choosing
j we have ensured that h (~;~X) n is represented by ~ manifolds of (genuine) dimension n and that hq( ;~) is represented by mock bundles of fibre dimension q. The first half of the theorem is true for non-
connected spectra by the same proof but since
X(X.) will in general l
have varying codimension in X., the cycles will be of mixed dimension, l
possibly of unbounded dimension. Examples 5.3.
1.
is just a framed manifold.
See the examples below.
X. = sj. l
X(X.) = pt. l
bordant to 0.
and an X manifold
1
i Z
2
144
=
pt. (of codim i), C
'
2
=
C(M)
is the union of a framed n-manifold
If f '" * then M is a framed manifold
see Example 6.4(4).
of bordism.
Similarly with ~
we get submanifolds
are 'virtual',
they are of the form
In this C(framed
and serve to allow the normal bundle of X(M, f) to be non-
trivial. .
5. 1
=
'+1
X = Moore spectrum. E. g. for Z , X. = SJ U DJ . n J n ¢, L = n points and we get framed Z manifolds. In n
2
general the geometric description fits with that given in Chapters III and IV (for stable homotopy, but see also Example 8 below).
v
If ~ and yare
TCW spectra then so is ~
v
y and an
Y manifold is merely the union of an ~ manifold with a y maniTake care about dimensions, 7.
L = 0, C 1 1 L = Mk-i-1
then the theory is a 'mixed
there is an elementary resolution connecting
example the singularities
fold.
Then
i<,
with normal vector bundles, i. e. smooth(able) submanifolds.
X
XZ+i = S Xi'
v
n
representation
Then L
< i,
sj
is a TCW, so is MPL and X(MPL) = BPL • Thus X(M, f) n n n n is just a submanifold of M of codimension n and we recover the usual
6.
X. = S\
=
BPL
for
For some j, Xi = Si u Dk where f : Sk-1 - Si is a given f
map and
J
X = Thorn spectrum ~ or !Y!9 etc. MPLn has a cell in which each cell is of the form (cell of BPL ) x 'in. Then if
4. structure
-
stable (co)homotopy. 2.
i. e. X.
Thus X theory is not a mixed dimension theory but it has
the two theories,
sphere)
Thus we have the usual representation
= *,
with a framed (n+j-k).,.manifold.
-------------
Remark 5. 2.
If f
dimension' theory, i. e. an 'n-cycle'
X(X.) to have constant codimension l
The case f "'"* of the example 2 is also worth discussing
3.
a resolvable singularity:
~ are equivalent to the bordism and mock bundle
theories based on stably framed
in the larger stratum is a product with C(M).
i, with the
obvious identifications given by the inclusions
Theorem 5. 1.
have two strata both framed and the neighbourhood of the smaller stratum
Each new cell attached determines a manifold in the previous theory,
j.
J
where M is the framed submanifold corresponding to f. ~ manifolds
product ~
If ~ and A
¥
are
see Example 3 above.
TCW spectra then so is a naive smash
y, see [1; p. 161], and a cell of ~
A
Y
is the product of one
of ~ with one of y and has for link the ioin of the corresponding links
145
for X and Y. Thus an X •....
Y. manifold is a manifold with singularities being joins of ~ singularities and y singularities.
The following theorem is proved by an argument similar to that
A
"",
8.
Combine Examples 4, 5 and 7. Then bordism with co-
contained in Chapter I (in fact rather simpler).
efficients has exactly the description given in part III!
Theorem 6. 1.
6. RESOLUTIONOF SINGULARITIESAND THE MAIN THEOREM Let T be a geometric homology theory and suppose that in T C(M)
(2)
C(C(M) U
Its proof will therefore
Let CW and SCW denote the categories of CW
complexes and isomorphisms onto subcomplexes and CW spectra and isomorphisms onto subspectra respectively.
Let
L denote the expansion
in CW or SCW. Then there are isomorphisms
there are two singularities of the form (1)
*
be omitted.
W) where W is a bordism (in T) of M to fi.
M Suppose also that C(M) does not appear in any other basic link.
hCW
£:;<
CW(L-1)
hSCW
£:;<
SCW(L-1).
Combining with 2.4 we have:
Then a T-manifold can be resolved so as to delete both of these two singularities.
Corollary 6. 2.
This is done by resolving all the C(M) type singularity using
There are equivalences of categories
(2) which is essentially a bordism in T of C(M) to a T-manifold with no C(M) singularity.
1
hCW "" TCW(L-
The method is similar to the proof of exactness
in IV Proposition 4. 1. Once there are no C(M) singularities, there can be none of the second type either.
) 1
hSCW "" STCW(L-
then
The precise description of
Now let
).
9 denote the category whose objects are geometric
this resolution process is contained in the CW interpretation which
theories and whose morphisms are inclusions of theories or 'relabellings'.
follows. Let T' be the theory in which these two singularities do not
We now regard all theories as theories of framed manifolds with singu-
appear then we say that T' is an elementary resolution of T. A
larities,
simultaneous family of elementary resolutions will also be called an
allowing singularities corresponding to the twisting of the normal bundle.
elementary resolution.
A theory T is included in a theory T' if the links of T' include those
A resolution is a countable sequence of elementary
resolutions.
non-framed manifolds being dealt with as in Example 5.3(4), by
of T up to a relabelling.
Now suppose that X and X' are TCW's and T and T' above
Let R denote the resolutions in g.
From the
discussion at the beginning of this section and 5. 1 we have an isomorphism
are the corresponding theories. Then X' C X and X differs from X' by the addition of two cells e n and e n+1. wIth en+1 attached by degree 1 on en and en otherwise free. an elementary collapse.
In other words X' differs from X by
Thus the geometric analogue of collapsing is
Combining this with 6.2 we have:
resolution and the process of resolution of a given T-manifold described
Main theorem 6. 3.
There is an equivalence of categories
at the beginning of the section is just transversely deforming the map g(R) "" hSCW .
Sn -+ X (which defines the manifold) into X', using the deformation retraction
X -+ X'.
The inclusion X'
C
X is an elementary expansion.
An expansion is a countable sequence of (simultaneous families of) ele-
*
mentary expansions.
Chapter I §4. These can be dealt with in a similar way.
146
There are similar set theoretic problems to those encountered in
147
In other words the stable homotopy category is equivalent to the category
is an operation and in fact many classical
of geometric_t!!f!0ries
(cf. McCrory (4)).
v
and
with resolutions_ formally inverted.
in hSCW are described
A
of singularities,
see
Remarks
mock theories
The operations
as 'union' and 'join'
5.3(6) and (7).
and corollaries
using 'embedded'
geometrically
geometric
of §4).
map ~
6. 4.
theories
1.
There is an unstable version,
(corresponding
We obtain an equivalence
to the embedded
theorl·es glOveequl·valent homology theories
3.
of uniqueness Two geometric
if and only if they differ by a
and their inverses.
The theorem
also describes
(i. e. natural transformations
of theories,
the stable maps geometrically operations
always has the form inclusion (relabelling)
etc.).
Such a map
followed by resolution.
follows from an analogue of Lemma 2. 1 of Chapter I.
4.
o
*.
of f to
are elementary U
Let X' = SJ
Make X' transverse
defines a geometric
C(C(M)
aW
U
resolutions.
in 5. 3(3) fits into the setting of this k (D x I) where H IS the homotopy
H relative
to the two 'ends'.
The new singularity
of the higher dimensional
This example makes it clear how resolutions
sj
l,
f
Dk
-+
given by collapsing
A
~'C
0
Now if ~n ...•~',
~'.
§n
A
By expanding
~ by a sequence
we can replace this by an equivalent
§m ...•~'
~m ...•~'
represents
represents
A~'
inclusion
two manifolds
their geometric
in
prod-
0
0
0
0
0
0
examples of rmg the ones
(bordlsm etc. ) are essentIally
example! . Note " however
the case of R-bordism
which the ring structure
appears
naturally
,
the general
see Chapter III , in
via a resolution
of singular i-
ties. REFERENCES FOR CHAPTER J. F. Adams. ChicagoU.P. [2]
Then X'
described
in 5.3(3)
in X' is
piece,
vn
Stable homotopy and generalised
homology.
(1974).
can change the appearance
P. Gabriel and M. Zisman.
Homotopy theory and the calculus
fractions.
Berlin (1967).
Springer-Verlag,
[3]
C. McCrory.
Cone complexes and duality,
[4]
C. McCrory.
Geometric
or conversely.
of a manifold drastically.
If we now consider
with suitable properties.
is one in which there is a
uct (see 5. 3 Example 7). Thus a theory has products if and only if it is possible to find a geometric representation in which the product of two o. cycles IS agam a cycle (after possIbly relabellmg). Thus the classIcal
pt. ) and can be used to resolve either the lower dimensional[5]
(and dimension)
••••~
have this form,
0
U
theory of which both the theories
piece into a singularity
5.
~'
[1] The example described
~
of mapping cylinders
This
See also the next
two examples.
section as follows.
A
Finally a theory with products
the theory then
ES (R) "" hCW.
2. Theorem 6. 3 gives the answer to the problem of geometrl·c representatl·ves for a given homology theory.
sequence of resolutions
6.
operations
objects, [6]
(to appear).
homology operations,
C. P. Rourke and B. J. Sanderson.
of
(to appear).
CW complexes
as geometric
(to appear).
D. A. Stone.
Stratified polyhedra.
Springer-Verlag
lecture
notes
No. 252.
the map
Sk sj to a point, then the geometric
description
of this k map is 'restrict to the singularity'. This is seen by including sj uf D j 1 k in D + U D which collapses to ~ and using an argument like f
Example 4.
This example makes it clear that 'restriction
to a singularity'
149
148
IIZ_!!!r!!l1I!ll!!,_m..
~.1iIIIII0'1III!.-.!If" -. ----------:-;
, .-:~- ••••••
w_-
-Jt_
