Brown-Peterson Homology: An Introduction and Sampler (Cbms Regional Conference Series in Mathematics 48)

BROWN-PETERSON HOMOLOGY: AN INTRODUCTION AND SAMPLER Conference Board of the Mathematical Sciences REGIONAL CONFEREN...

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BROWN-PETERSON HOMOLOGY: AN INTRODUCTION AND SAMPLER

Conference Board of the Mathematical Sciences

REGIONAL CONFERENCE SERIES IN MATHEMATICS supported by the National Science Foundation

Number

48

BROWN-PETERSON HOMOLOGY: AN INTRODUCTION AND SAMPLER

by

W. STEPHEN WILSON

Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island

Expository Lectures from the CBMS Regional Conference held at the State University of New York, Center at Albany 2000

June 23 27, 1980

Mathematics Subject Classification. Primary

55-XX.

Library of Congress Cataloging-in-Publication Data Wilson, W. Stephen, 1946Brown-Peterson homology. ( Regional conference series in mathematics; no. 48) "Expository lectures from the CBMS regional conference held at the State University of New York, Center at Albany, June 23-27, 1980"-T.p. verso. Includes bibliographical references. 1. Homology theory. 2. Homotopy theory. I. Conference Board of the Mathematical Sciences.

II. Title.

III. Series.

QA1.R33 no. 48 [QA612.3] ISBN 0-8218-1699-3 ISSN 0160-7642

510s

[514'.23]

81-20619 AACR2

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissiont!lams. org. Copyright

© 1982

by the American Mathematical Society

The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6

54 3 2

10 09 08 07 06 05

Contents Introduction.......... . . . . . . . . . . .. ............................................................. ....................................................

1

Part I. An Introduction. Section

1.

Section

2.

Section

3.

Complex bordism..........................................................................................

3

Formal groups...................................................... ..........................................

12

Brown-Peterson homology............................. ..............................................

14

Part II. A Sampler. Cooperations an d stable homotopy.............................. ............ ............... . . .

23

Section

4. 5.

Associated homology theories.................................... .................................

34

Section

6.

Morava's little structure theorem and t he Conner-Floyd

Section

Section

7.

conjecture....... ..................................................................................................

37

K(n).K...............................

4

and the Stee nrod algebra..................................... .............................

51

Hop f rings, the bar spectral sequence, and

Section 8.

H*K*

Section

9.

Two formal groups and

Section

1 0.

Chan's proof of no torsion in

BP.BP................................................................... . ...

.

. ................... ........................... ... .

.

.

56 59

Part III. Something New. Sectio n

11.

Unstable operations............................................................................... . . . . ... . .. .

References................. ...................... ............ .......................................................................................

v

61 83

Introduction A few years ago, I hoped to have the opportunity to write lecture notes such as these . I t was then possible t o do a complete survey o f the post-Stong

[St]

research i n t he direction

of Brown-Peterson homology. When fmally confronte d with this outlet,

I

found it impossi

ble to do justice to the e ntire field. Although grand schemes such as another book like Stong's or an exposition of Morava's work did occur to me ,

I

had only three months to pre

pare the lectures. The nature of C. B. M . S. Regional Confe rences dictated that the lectures should expose nonspecialists to the a rea (although few were present). The format became an introduction to the basic facts about BP, with proofs, and a highly personal sampler of fur ther material, mostly without proofs. These notes are fairly t rue to the lectures;

much.

(I have

I

talked too

adde d §8 and expan ded § 11.) I have tried to retain an informal style and

where clarity means lying , the notes are clear. An introduction to

BP is

a construction of

BP

and a study of the stable operations. In

the sampler I discuss a number of distinc t tools: operations, associated homology theories, formal groups, and Hopf rings; and show how each applies i n several ways to interac t with the others. Throughout , I have emphasized the transition from sophisticated internal theo rems to the necessary applications; the moral: support basic research . The Conner-Floyd con jecture emerges as a central figure in these notes; it depends on almost everything else and therefore is difficul t to discuss outside of a series of lectures like these . A diagram of de pendencies for the Conner-Floyd conjecture is given as: Hopf rings

Morava structure theorem

�

bar spectral sequence

\ I /

Morava K-theories of Eile nberg-Mac Lane

Conne r-Floyd conjecture

Each item in the diagram has other uses as well. Also, since these lectures, serious compu tations for

BP X *

have been done which de pend on the Conner-Floyd conjecture .

2

W. STEPHEN WILSON

I begin by assuming the reader is familiar with the homology and homotopy of the spectrum MU. From this I develop the basic facts about complex bordism cooperations (§ 1), introduce the minimal necessary formal group machinery (§2), construct BP (§3), and describe its cooperations (§3). The sampler is begun by going into more depth on the cooperations and applying them to stable homotopy (§4). The algebra discussed arises again and again in later sections. Next , I introduce the sequences of associated homology theories which contribute so much to the rich and plentiful internal theorems in the subject (§5). Operations and associated homology theo ries are combined to describe Morava's little structure theorem ( § 6). A little knowledge of the Morava K-theories of Eilenberg-MacLane spaces is fed into the Morava structure theorem to prove the Conner-Floyd conjecture (§6). To obtain this "little knowledge" it is necessary to study 0-spectra, g•. The main tool is Hopf rings and they are inserted into the bar spectral sequence (§ 7). In order to complete the proof of the Conner-Floyd conjecture, the first appli cation is to the Morava K-theories of Eilenberg-MacLane spaces (§ 7). As a detailed example of Hopf rings and the bar spectral sequence at work, the homology of the Eilenberg-MacLane spaces is computed {§8). Hopf rings and formal groups are then combined to stutiy the n spectrum for BP (§9). This provides the opportunity to give Chan's proof of nc� torsion (§ 1 0) and leads naturally to the study of unstable operations (§ 1 1 ). The material on unstable opera tions is previously unpublished. For an introduction, see the beginning of § 1 1 . Although there are many new ways to do the material, § § 1 -3 follow Adams [A2 ] The required properties are established as quickly as possible; the student and researcher alike will need to read Adams's book for the wealth of results which do not lie on the geo desic to BP. Mike Boardman made the inclusion of the Hazewinkel generators possible by supplying the simple proof used here . § §4- 1 0 owe debts to all of the coauthors and colleagues mentioned in the text. §8 is new and is joint work with Douglas C. Ravenel. The acknowledgements for § 1 1 are in the introduction to that section. The idea of these lectures is entirely due to Tim Lance. I offer my warm thanks to him for this opportunity and also to Mike Chisholm and the other members of the faculty of State University of New York at Albany who helped him out. Thanks are also due the Conference Board of the Mathematical Sciences and the National Science Foundation (who funded the con ference and have cooperated with my tardiness), and the participants of the conference. I thank the Osaka City University, Japan, the Tata Institute of Fundamental Research, India, and the Hebrew University, Israel, for their hospitality and support during the produc tion of these notes, and Mrs. M. A. Einstein at The Johns Hopkins University for difficult mail-order typing. During this period I was also partially supported by the National Science Foundation and the Sloan Foundation. My apologies to the reader for not including a complete bibliography of the subject, but I found it too difficult to attempt away from my own files. Finaly I want to thank Michael Boardman, David Johnson and Robert Stong for numerous comments on the manuscript which alowed me to correct errors and improve the •

exposition.

Part I. An Introduction 1. Complex bordism.

Bordism begins with geometry. Manifolds are well-studied objects; bordism exploits the reservoir of knowledge about manifolds to attack homotopy theory for more general spaces. This exploitation is analogous to the use of the geometry of Euclidean cells by standard homology . In classical homology we quickly subordinate the geometry to the rich algebraic structure which it induces. Likewise with bordism. We must, however, keep the geometry in mind, not just for philosophical reasons, but because we often return to bordism's geometrical roots to derive new algebra. For the correct details for the first part of this section we suggest Stong's book [St] ; for the second part, Adams's book [A2 ] . Let v

l

(1.1)

be the stable normal bundle for a manifold Mn of dimension n. Then v is the pull-back of the universal bundle � over BO by a map f (which is unique up to homotopy): v

(1.2)

--

�

l 1

Mn�so f The inclusion U(n)� 0(2n) induces the map

(1 .3)

r:

BU�BO.

This map pulls back the universal real bundle to the universal unitary bundle. If we can choose a homotopy lifting g,

(1.4)

3

4

W. STEPHEN WILSON

we say that Mn has a U-structure g. It can have several U-structures or none. A pair (Mn, g) of a manifold and a U-structure is called a U-mani[old. There is also an "opposite" U-struc ture on Mn, (-Mn, g) (cf. [CF3]). We define the nth complex bordism group of a space X, MUnX. We consider all triples (Mn , g, f ), where (Mn, g) is a U-manifold of dimension n and f is a map ( 1 .5) As with chains in homology, there are far too many of these to take seriously the first time around. (They can, however, be useful.) So we put an equivalence relation on these triples 1 by (Mf, gl' [1)- (M�, g2, [2) if there is a triple (wn+ , h, k) with o(wn+ 1, h) = (Mf, g 1 ) 1 u -(M�. g2) and klawn+ =[1 Uf2: Mf

(1.6)

wn + 1

�X

,

awn+1

=

Mf

U

-M�.

M�

Under this equivalence relation we have an abelian group MUnX. where addition is given by disjoint union. Using manifolds with boundary one can define bordism for a pair (X, A) and verify geometrically that MU*(X, A) is a generalized homology theory [CFl' CF3, C]. Generalized homology theories like this are always obtained from spectra [Br, Wh] . The spectrum for complex bordism is just the Th.om spectrum for the unitary groups: 2 MU = {MU(n)}n>o with maps S /\ MU(n - 1 )- MU(n). ( 1 .7) The space MU(n) is the Thorn space of the universal n-dimensional complex bundle �n over BU(n). The maps are induced by the maps �n-1 ec

1

( 1.8)

BU(n -

-

�n

�

I ) - BU(n)

We have an isomorphism: (1.9)

MUn X � lim [sn+lk, MU(k) I\ X] = {Sn, MU I\ X}= {S0, MU I\ X} . n k-+oo

(We are making free use of Boardman's stable homotopy category [V] . ) This isomorphism is proven using transversality as in Thorn's original paper [T] ; see [CF 1, CF3, C]. We can use the right-hand side of (1 .9) to define complex bordism. We see that all of the informa tion of complex bordism is contained in the spectrum MU. (We have now reduced to homo topy theory the geometry we wish to use to study homotopy theory to solve the geometric problems which have been reduced to homotopy theory.) To study bordism we need to know about MU. We will ignore the geometry as we develop the basic properties of MU.

BROWN-PETERSON HOMOLOGY

5

Where there is a generalized homology theory, there is a generalized cohomology theory. In this case we have complex cobordism defined for finite complexes by

(1.10)

2 Munx = lim [S k-nx, MU(k) ] = {s-nX, MU} ={X, MU}n . k-+oo

The Whitney sum of bundles gives a map

(1.11)

x

BU(n)

BU(m)--+ BU(n

+

m),

+

m).

which, when we pass to the Thorn spaces, gives

(1.12)

MU(n) 1\ MU(m)--+ MU(n

This pairing turns the spectrum MU into a ring spectrum, (1.13)

m: MU 1\MU--+MU. This makes MU*X an algebra over MU* =MU* (point) and MU*X a module over 1r*MU = MU* = MU*(point) = MU-* . This algebra follows from the geometric Cartesian product of U-manifolds. The ring structure tu rns

(1.14)

H*MU = lim H*+lkMU(k) k-+�

into an algebra with a simple description. We combine the homotopy equivalence MU(l) ::.: CPoo and the map used in defining the spectrum: ( 1 . 1 5)

x:

CPOO

::.:

MU(l)--+ MU.

Applying homology we obtain a map

(1.16) TI1e group H2;+ 2CPoo is free on one generator denoted by pi+ 1• We define ( 1.1 7)

to be the image of pi+ 1 under this map. We have

(1.18) a polynomial algebra on the b's. Since these generators, b;, come from C/'00, the comodule structure over the dual of the Steenrod algebra can be computed. Using this, the homotopy of MU can be computed with the Adams spectral sequence to obtain

(1.19) a polynomial algebra on even degree generators. An element x2n E 1r2nMU C H2nMU is a where A = ±p if n = p ; - l for some prime p and polynomial generator if x2n = Abn + A = ± l otherwise. We assume that the reader is familiar with these basic homology and homotopy facts about MU. They are readily accessible. See Stong's book [St], Switzer's book [Sw] , or the original papers of Milnor [Mi1] and Novikov [N}. ·

· ·

W.

6

STEPHEN

WILSON

We can consider the Hurewicz homomorphism ( 1 .20) The groups MU* are just the equivalence classes of U-manifolds under the bordism relation ( 1 .6). Denote the equivalence class of CP" with the standard U-structure by [CP"). We have (1.2 1) These elements form rational generators. We need to identify these elements in homology. This is one of the key connections between the geometry and the algebra. In the power series ring H.MU [ [s]), define ( 1 .22)

exp s= 2:

;;;.o

b;si+ 1•

b0 = 1 .

Define ( 1 .23)

log s=

L mnsn+t, n;;.o

by

( 1 .24)

log(exp(s))= s= exp(log(s)).

The mn (n > 0) are new polynomial generators for dix I],

H.Mu and from

Miscenko [N, Appen

( 1 .25) This is proven using characteristic number arguments and is part of the assumed knowledge of the homology and homotopy of MU. * We see that the Atiyah-Hirzebruch spectral sequences for MU.CP""' and MU CP""' both collapse (they are in even degrees). Thus ( 1 .26) where xEMU2CP""' is the map ( 1 . 1 5) (recall ( 1 . 1 0)). This all follows from the compati bility of the spectral sequence with the usual pairing of (generalized) homology and coho mology. We also need the obvious fact that x reduces to the standard cohomology algebra generator. To describe the Kronecker pairing above, let * ( 1 .27) y: S-+MU AXEMU*X and z: X-+MUEMU X. Then

(z, y)EMU*

( 1 .28) is ( 1 .29)

S 4MU 1\ X

Il\z

MU 1\ MU..!MU

BROWN-PETERSON HOMOLOGY

7

(good for any ring spectrum) where m is (1.13). Deflne

(1.30) by ( 1.31) This definition is valid because of the similar duality for homology and the compatibility of the spectral sequence with the pairing. These {3's reduce to the standard homology (3's under the map ( 1.3 2) which we have from our knowledge of H0MU. Using our map (I.IS), we define (1.33) just as with homology (1 .1 7). The Atiyah-Hirzebruch spectral sequence (1.3 4) collapses (it is even degree), giving {1.35) {The upper MU on bJ'fu will appear only when it seems necessary.) This is the continuous * dual of the Landweber-Novikov operations MU MU [L1, N]. {The topology is obtained from the finite skeleta of MU.) Using {1.36) to get the map {1.37)

x � s0 A x�Mu Ax,

we can apply MU* (-) to get (1.38) We need a standard lemma to proceed. LEMMA 1.39. Let E* (-) be a reduced generalized homology theory. Let E*X be free over E*. The exterior product gives a Kunneth isomorphism: o E*X ®E • E*Y-=-+ E*(X A Y). PROOF. Both E*X ®E • E* (-) and E*(X A -) are generalized homology theories .

The exterior product provides a natural transformation between them. They agree on a point so the E2 terms of the Atiyah-Hirzebruch spectral sequences are isomorphic and we are done. o

8

W.

STEPHEN WILSON

By ( 1 .3 8) and Lemma 1 .39 we have ( 1 .40) MU*MU = tr*(MU 1\ MU) has two distinct MU* module structures: a left and a right (see below). From ( 1 .35) we know the left module structure is free; so, by symmetry, the right is free. In the tensor product of ( 1 .40) we are using the right module structure. Let X = MU and we see that MU*MU is a ••Hopf algebra". For a general X, MU*X is a comodule over this Hopf algebra. We know the algebra structure of MU*MU already from ( 1 .35). We give the other structure maps; see [A1]. We have ( 1 .4 1 ) just apply 1r *(-) to (1 .13). The left and right units, ( 1 .42) are obtained by applying ( 1 .43)

tr *

MU 1\ S0

( ) to

--

S0 A MU-- MU 1\ MU

MU 1\ MU and

respectively. Next we have the conjugation ( 1 .44)

c:MU*MU which comes in the same way from the switch map (1.45)

�.

MU A MU

We consider MU* C H*MU, ( 1 .20), and describe our maps accordingly. Note also that ( 1 .46) Use bf" u to define exp and log series. As with homology, we obtain m f" U; see ( 1 .22)-(1 .24). At times we write m MU

( 1 .47)

b

=

=

i�O

mI!'-fU '

L b;. etc.

i�O

When we use these symbols in equations, we mean the equation is valid degree by degree.

THEOREM 1 .48. (a) e(b;) = 0, i > 0, i.e. e(b) = 1 . (b) 11L(m) = m and 11R(m) = '1;i�om;(mMU)i+1. 2 (c) c = J, C'T/L = 'TIR• C'T]R = 'TIL• and c(bMU) = mMU. (d) The coproduct (1.40) is = '1;i�0bi+ 1 ® bi" (e) The coproduct for MU* CP"" is 1/JC ""(�) = '1;i�0bi

P

® �i·

o

9

BROWN-PETERSON HOMOLOGY

These are formulas which we shall use later to obtain the corresponding formulas for BP. Because this is an introduction to the subj ect we will give the details of the proofs. Part (a). e(b;) = .o. b;> = (1, x*pi+ 1) = (x *(l), Pi+ 1) = <x, /3i+1) = 0, i > 0. The remaining formulas are a corollary of one result. By m ap pin g MU-+ MU A MU in to the left or right factor as in (1.43) we can define elements xL, xR, f3f, pf in (MU A MU)*CPoo � (MU A MU)* [ [x] ]

( 1.49)

and (MU A MU)*CP ...

respectively. Here

(MU A MU)-*

(1.50)

LEMMA 1.51. xR Before we begin

=

=

(MU A MU)* = 1r*(MU A MU) = MU.MU. 'i:,;;;.obf'1U(xL)i+1. o

th e proof,

we need:

LEMM A 1.52. The following diagram commutes:

and P is an isomorphism. Here R is the Boardman map induced by MU =

S0

A

MU-+ MU

A MU,

a:(f )(f3;)

=

f*(IJ;).

and

where { , ) is defined because MU A MU is an MU module spectrum by the map o

MU A (MU A MU) � (MU A MU) A MU � MU A MU.

PROOF. rl(f)(f3;) isS -+fJ; MU A CP"'" -+1At MU A MU. R R(f) is CP"'" -+I MU � SO A MU-+ MU A MU. So, P(R(f))(f3;) = is P

S�MU 1\ CP""

-4 MU A !tfU � MU A S0 A MU IAR MU A MU A MU .... ..

--- ------- - - ---

mM

MU A MU.

_."'(

The last maps can be ignored because they give the identity. We see that we are left with rl(f)(/3;). thus proving the diagram commutes.

The fact that P is an isomorphism is really the more general fact that F *CP .. � homE • (E.CP .. , F*) when F is a module spectrum over E and the Atiyah-Hirzebruch spectral sequence collapses. o

W. STEPHEN WILSON

lO PROOF OF 1.51.

We have

a{x)(/3i+ 1)

P(xL)i(/3;)

=

bfu by

=

<(xL)i,/3 > ;

=

� MU 1\

MU 1\ MU

CP""

of.

<(xL)i,/3;>:

To show this, we write down the map for the element

S

definition . We also have

1\ MU

m/II

MU 1\

MU

an d observe that it factors

S � MU 1\. CP"" IA (xL)i

MU 1\

l

MU 1\ MU � MU

!Axi

1\ MU

r

MU 1\ MU --=- MU 1\ MU 1\ S 0 � MU 1\ so

This is alre ady

o{ in

the lower

corn e r. Since

(MU 1\ MU)*CP"" � (MU 1\ MU)*[[xL ] ] ,

a; = =

(

1.51 to

obtain

j �O

P(R(x))(l3i+1)

We return to the

Part (c).

L a;(XL )i+

proof of

a:

=

<xR. 13i+1> = P(xR)(I3i+ 1)

(x)(/3; + 1 )

= bfu.

o

The first three parts are obvious from definitions.

71L(m)

= m

=

1.51

c

to

L c(bfu)(xRi+ t.

i�O

which define s

integra l Eilenberg-MacLane spectrum.

two formulas for xH:

For the la st, apply

c(b{"f u) = mfu. follows because 11 is a left module map. For 11 • le t H be the L R L

Thus it is the inverse series to (b).

=

)

13i+ 1

1.48.

xL

Part

1,

the m{"fu

so we have

We reduce the formula for

x

as follows and obtain

11

BROWN-PETERSON HOMOLOGY So

or

(mMU)i+ 1 .

'TIR (m) _

Part

(e). The coproduct for

CPoo is

defined by

®Mu. MU.CP" (MU The element in terms of

1\

MU).CP""

(11 goes t o [Jf, i.e . 1/lcp(f31)= fJf, and 1 ® .(11 is just {Jf. We need R = �;>obflu(xL)i+1 = ��= a1_k ® pk. We recall x 0

{Jf, say {Jf

(xR)i where

(bMU)'

Part (d).

means the

2k degree

=

L (bMU),(xL)i+k

component of

This follows at once from part (e).

MU.

(bMU)i.

We have

o

LEMMA 1 .53 . There is a 1- 1 co"espondence between maps MU--+ MU, and power series

f(x)= Ld1xi+l EMU2CPoo, I>O

by g.(x) = f(x).

d0

=

of ring spectra, g: 1.

o

PROOF.

MU*MU�homMu.(MU.MU, MU.), MU*(MU /\ MU)� homMu (MU.(MU /\ MU), MU.), •

MU.(MU

/\ MU) � MU.MU ®Mu •

MU .MU.

{Jf

which gives

k>O

We will nee d one more result about

The co"espondence is given

to write

12 Let

W. STEPHEN WILSON

8� E homM u. (MU.MU. MU ) correspond to g � MU*MU. g MU 1\ MU Ag MU A MU .

The diagram

MU--g�-MU

8g is a map of algebras if and only if g is a map of ring spectra. d1 EMU.w i > 0 (8g(b0 1) 1), and let g.(x) l:1;;.0a1x1+1•

shows that

=

d;

=

=

Let 8 g{b;)

=

= 8g(b1) = 8g (x*{3i+1) = ( g, x*{3i+1> = <x *g, /3i+1)

)

1 L a1x i + , {3i+1

= =

=

a1•

o

2.

Formal groups. A great time can be had with formal groups, es p c d all y in connec tion with MU and BP. In fact it could easily be made the main theme of lecture notes with the same title as these. However, our theme must be elsewhere a nd the reader will regretta bly be exposed only to the necessary minimum. A commutative formal group law over a graded ring R is a power series

(2.1 )

F(x, y)

=

1 L.a 11x y i eR.[[x, y)], i ,j

* a11 ER2(i+/-l )•

(2.2} The .cohomology theory MU*(-) gives rise to a formal group law over apply MU*(-) to the standard map JJ.=

(2.3)

CP"" X

CP""

to get

(2. 4)

JJ.

*: MU*CP""

-+

-

MU..

Just

CP""

MU*(CP"" CP"") MU*CP"" � MU*CP"". x

=:!

We obtain the formal group law (2.5)

F(x1,

x2) = JJ. *(x) =:= 'L.a11x� � x�. i,j

Its properties follow from the properties of JJ.: homotopy commutativity, associativity, etc. As usual we can best describe F by considering

{2. 6)

MU*CP"" c (H ·A MU)*CP ·

""

.

BROWN-PETERSON HOMOLOGY

13

Applying c to 1.51 we have xL

(2.7) Reducing to (H

=

E m/"u(xRy+t E (MU 1\ MU)2cr0•

f>O

1\ MU)*CP"" we have x"

(2.8 )

=

1:; m;xMU

;;;.o

=

log x.

* The element x" is primitive, so apply p. : (2.9) Now apply exp(-) to obtain: (2.10) LEMMA 2.11.

The primitives of MUQCP""' are MUQ free on log xMu.

o

PROOF. Because H*(CP"", Q) is a polynomial algebra on a. two dimensional element,

MU�CP""

is a polynomial algebra over MU$ on a two dimensional element. By duality * there is only one primitive free over MU rationally. We know log xMU is a primitive' and a quick check of its leading coefficient shows it is the generator we want. o The ring for a universal formal group law can be constructed using arbitrary a;/S and the relations from (2.1). Lazard has determined the structure of this ring. The theory of rational formal group laws is trivial; (2.10)-(2.12). One of the strongest connections between bordism and formal groups is displayed in the following theorem. THEOREM 2.12 ( QUILLEN [Q2), OR SEE (A2] ). The formal group law for complex cobordism is isomorphic to the universal formal group law of Lazard. o Given any formal group law F(x, y) over R*, there is a ring map g: MU*-+ R* which induces the formal group; that is (2.13) The result can be made to appear even stronger because a topological analogue is true. It follows, however, from a sliht generalization of 1.53, even without Quillen's result, and it does not imply 2.12. THEOREM 2.14. If F(x, y) is a formal group law over E* which comes from a gener * alized cohomology theory E (-) and the map (2.3), then there is a map of ring spectra, MU-+ E, which induces the formal group law. o We define the formal group sum: (2.15)

x

+p y

=

F(x, y)

=

exp(log x +logy).

W.

14

STEPHEN WILSON

We defme [ l](x)

(2. 1 6)

=

x

=

e xp(log x),

and, inductively, [n](x) = [n- 1](x)

+p

x

=

e xp( n log

x),

n > l.

We have [-l](x)

(2. 17)

=

t(x)

= e xp(-log x)

which implies

F(x, t(x )) = 0.

(2. 1 8) To prove this is well defined, let

t(x)

(2.19)

=

L ckxk +t. k>O

Then (2.20) can be solved inductively for en since a0 1 = 1. This allows us to do formal group subtraction: X

(2.2 1 )

-F y

= X +p

((-l](y)).

We need only one more elementary formal group fact for the construction of BP. LEMMA 2.22. The power series [1/d] (x)

= ex p(l/d log x) is defined over MU* ®Zr 1141•

PRooF. Clearly,

[d]([1/d] (x)) Let [ 1 /d] (x ) =

l:,i >Ocixi +t

=

x))) = exp(d 1/d log x) = x. l:,i>Oaixi+1 ; then

e xp(d log(e xp(l/d log

and

[d}(x) =

can be solved inductively for en. We get that c0 = a01 = 1 /d and en is obtained in terms of a1, ci , j < n, and a01 ; so when d has been inverted, the power series [1/d] (x) is de fmed. o 3. Brown-Peterson homology. Brown and Peterson constructed BP, known to them as X, in [BP) . They built BP using a generalized Postnikov system. Novikov also gave a construction [N] . BP was not computationally useful. until a description of its operation ring was available. This was produced by Quilen [Q1] and written up by Adams [A2] •

15

BROWN-PETERSON HOMOLOGY

MU at a prime p and find a multiplicative idempotent e of MU(p)· The image of a multiplicative idempoten t in MU*(X)(p) is a natural direct summand We localize the spectrum

and so gives rise to the multiplicative generalized homology theory called Brown -Peterson

homology:

BP*X.

The representing spectrum is called BP.

is that there are no pointwise models for

BP,

One of the problems in the field

only homotopy construction s .

The ide mp otent o f interest i s the composition o f a family o f commuting idempotents . W e describe these ide mpotents.

LEMMA 3.1. Let q be a prime. There exists a multiplicative idempotent MU1 11q 1 such that on H*(MU1 11q 1)

eq on

n + I =F 0 (q), n+ 1 The

eq

commute.

=

0 (q).

o

From this we have:

THEOREM 3.2 ( QUILLEN [Q1) ). The multiplicative idempotent

= fl eq E MU *MU(p)

e

q#'-p

n=l=p i-1, n =pi- 1. The image of e* in MU*(X)(p) is a multiplicative generalized homology theory denoted by BP*X, with

PROOF OF

3.2.

The product

because for large primes

€ an ide mpoten t .

eq

llq,;.p eq

MU*MU(p) The commuting of the eq makes

is c onvergent in the topology on

is the identity for large ske leta.

The rest follows from the lemma and above .

o

Lemma I .53 on multiplicative maps is true after localization . The same proof applies .

Multiplicative maps are identified there in terms of what they do on is describe d in terms of what it d oes to formation to the other type . Let operation

(3.3)

g.

H*MU1 11q 1 .

f(xMU) x

=

Our

eq , h owever,

be a p ower series representing a multiplicative

We define mog

xMu.

We need to go from one type of in

L g*(mn )xn + 1 . n ;;.o

W. STEPHEN WILSON

16

Since log xM U is primitive, (2. 1 1), by chec king the first coefficient and using the fact that g*(Iog xM U) must still be primitive, we h ave (3.4) Also, (3.5) Letr1(x) be the inverse power series, i.e.[-1(f(x)) = x = f(f 1 (x )) . From (3 .4)-(3.5) log x M U = mogf(x M u),

{3.6) so

logf-1(x) = mog x

{3.7)

and by applying exp(-) we have (3.8) f-1 ( x) = exp(mog x). We have computedf(x) in terms of g*(mn) (and vice versa). PROOF OF 3. 1 . We have a chosen mog for eq. Our only problem is to see that the correspondingf(x) has coefficients in MU* after inverting q. This is done indirectly. Let �; q be the qth roots of unity, q � i > 0. Since x - I = ll1>0(x-�;), most symmetric func tions are zero. In particular,

k =F 0 (q),

. �� + .. +�! = We see

tnat

q,

k

=

0 (q).

we can rewrite our desired mog x as

mog x= log x- .!(log � x +···+log�q x ) . 1 q Our formal group nonsense comes in handy here. This tells us, by (3.8), f-1(x) = exp(mog X) = X F -

[�]�< 1x

+p .

· ·

+p�q x)

.

Because this is obtained using formal group sums and the [1/q] sequence, we see that the coefficients off-1(x) are in MU*[� , ,�qJ llfqJ. However, all terms involving the�; 1 are symmetric, and the symmetric functions in the �; are integral. Thus [-1(x) has coeffi cients in MU*[ l/q 1 and since its first term is x, the inverse f(x) exists and has coefficients • • •

inMU*ltfqJ·

This gives a multiplicative map with the desired properties. The commuting of the eq can be checked by evaluating in homM u. [ lfq 1 (MU*MU1 l/q 1, MU *[ lfq 1) which is, ration ally, the same as homQ(H*(MU; Q), H*(MU; Q)). Li ke wise for eq o eq = eq. o We develop the basic properties of BP *BP. Just as with MU, we wil do our co mp u tati ons with BP*BP C (H 1\ BP)*BP.

(3.9)

We occasionally change the indexing on the m's so that (3. 1 0)

BROWN-PETERSON HOMOLOGY THEOREM 3.11 (QUILLEN

TJR(mk) = };r=o m;tk�;(ii) BP*BP�BP*[t 1, t 2 , (iii) c is given by

[Q1]). (i) There are • • •

t ;

17

E BP2(p;_l)BP, t0

= 1, such that

].

(iv) The counit e has e { l) = l , e(t;) = 0, i > 0. (v) The coproduct t/J is computed by k

.

L m;(t/J(tk-;))P'

=

F L t/J(t;)

=

i=O

L m tfh ® tl +i h+i+j=k h

or

i�O

F }_ t; ®

i,j�O

tf i ·

0

Ravenel first showed us the nice formal group sum versions of these formulas. Note the similarities with the formulas for the dual of the Steenrod algebra where formal group addition is just regular addition and you use the conjugates of the normal generators for the dual of the Steenrod algebra. PROOF OF 3.11. Parts (i) and (ii). The formula in (i) can be used to define the t's. The question is whether or not they actually lie in BP*BP, since they were defined in (H 1\ BP)*BP. We recall 1.48(b) for MU: 17R (m)

=

L m;(mMU)i+t.

i�O

Apply the idempotent of 3.2 to obtain L;�o TJR(mP;_1) = the left-hand side using the formula defining the t's in (i):

'l;; � o mp

BP ; i - 1 (m )P . Rewrite

Using these two expressions we have Lk�O logBPt k = logB P mBP. Apply expBP(-) to obtain (3.12)

Since mBP E BP*BP it is simple to prove, by induction on k, that tk is in BP*BP. Further more, it shows that tk = mpBkP-1 modulo lower m's. Since these m's reduce to the generating m's in homology we have part (ii). Part (iii). Apply c to (i),

W. STEPHEN WILSON

18 to obtain

mk =

; 11R (m;)c(ti) P ,

L

i+j=k

which is =

Add these over k and apply exp to get the formal group sum version of the formula. Part (iv). The proof is by induction . Apply e to (i), 11R (mk) =

L

i+j=k

; m;tf ,

to get mk = mk + e(tk), so e(tk) = 0. ; Part (v). Apply 1/1 to part (i), 'TIR (mk) = 'Li+i=km;tf • to get 1 ® 11R (mk) =

L

i+j=k

; m;l/l(ti)P .

The left-hand side, by part (i), is 1®

i+j=k

i m.tP If =

i+j=k

11R (m.J ) ® tjP

;

>

and again by part (i), =

This proves the first formula. Just add over k to get

L log 1/l(ti) = L

j�O

s,j�O

(.

log t8 ® t

Apply e xp to obtain the formal group sum version. o With our new generators t;. it seems that we have lost track of the coproduct on CP"". It can be recovered . This is an example of how much information remains in [A2}. Several people have studied operations on BP *CP"" without noticing the following simple formula (which, unfortunately for the author, was first conjectured by computations). given

THEOREM 3.13 [RW.J. The coaction 1/ICP"': BP*CP""-+ BP*BP ®8p BP*CP"" is *

by

PRooF. Apply the idempotent to the MU* coproduct 1.48(e) to obtain

"' .. (13BP) = L (bBP)i ® 13fP. CP i�O

BROWN-PETERSON HOMOLOGY

19

Substitute bBP = c(m BP) from 1 .48(c) and m BP = I; :;;. otn from (3 .12). For future use we record the formula used in the proof above .

o

0

LEMMA 3 . 1 4. bBP = c(I;:;;.otn )·

Even the most naive reader should realize there are problems with viewing BP* only as a subring of H.BP. Historically, general computations with BP had to wait not only for Quillen's description of BP.BP, but also for explicit generators for BP•· In retrospect, many results do not need explicit generators to be proven but they were essential for the first proofs. Liulevicius constructed generators for BP at p = 2 in (Li] . Then Hazewinkel con structed generators at all primes [H1, H ]. Other generators followed; [K., Ar1], etc. Un 2 fortunately, Hazewinkel's generators came after Adams's lecture notes [�] and were not included. THEOREM 3.15 (HAZEWINKEL [H ,

H2]). Generators [or

1 tr*BP c H*BP

z(p) [vi. v2

•

t�] z(p)t�[m . m , . . .] . . •

are given by

c

l

2

n-1

1 L miv: -1. o

vn = pmn

1=1

PROOF. From (1. 1 9) and the properties of the idempotent, it is clear that an element wn E BP2(pn - ) is a generator if and only if wn = Amn + P I A A =F 0 (p2). Our form t ula for vn meets this requirement if vn lies in BP•· · · ·,

LEMMA 3 . 1 6. exp(px) EpBP.[[x]).

C

,

o

n n PROOF. From ( 1 . 2 5 ) we know that p mn = [CPP _1] EBP•. Thus p kH 2 k(p -l)BP n tr k(p-l)BP, the worst case being m � , so p bn is clearly in BP•. Thus 2 exp(px) =

L b1(px'j+1 EpBP.([x]] . o

1">0

Rewrite the formula for vn as pmn

pm -p =

Write

p

=

I;f,;J m1 v:�t and add over all n

> 0:

L m1v( , O
as log(exp(p)). The above is now p log( I ) = log(exp(p))

(p](l}

=

F exp(p) +p L v . i>O

1

+ I;/>O log vr Apply ex p:

W.

STEPHEN WILSON

Since exp(p) EpBP * and [ p](I) has coefficients in BP* we can easily show, by induction onj, that vi EBP.. o In the above proof we showed: LEMMA 3.17. [p](x) =

"L{>0 vixpi mod (p).

o

This is a useful formula. Among other things, it says that the coefficients of xP" in the [p] -sequence are generators. This was surely known to Morava and probably Quillen be fore other people were even interested. We see , however, that the result can be obtained from [CF1], which existed before the question could be formulated. Araki's generators [Ard have the property that the formula in 3.17 holds without going mod (p). We have introduced the basic formulas for BP. The reader is now equipped to use BP as a tool. The formulas, however, can be difficult to use. They are inductive . In principle, one needs to know everything in lower dimensions in order to compute the next. This is in sharp contrast to the closed formulas in the Adem relations. In practice it is impossible to keep track of all of the information forever. Consequently, much of the j<'b of a beeper (BP'er) is to be able to skim some information from the formulas without having to deal with it all and come out alive . There is strong motivation to do this. The formulas contain a vast amount of information. The Brown-Peterson spectrum sits midway between the sphere spectrum and the Eilenberg-MacLane spectrum. It has a nice balance between its homotopy and homology:

homology:

good

okay

bad

homotopy:

bad

okay

good

It can be expected to be useful because a tremendous amount of information has already been used in its construction. It has the advantage of small size over MU. Anyone who has worked with a polyno mial algebra prefers one whose generators go up exponentially in degree Oike BP) to one where they go up linearly Oike MU). The mod (p) cohomology of BP is j ust the reduced pth powers, i.e ., the Steenrod aJ. gebra modulo the left-right ideal generated by the Bockstein, or the left ideal generated by the Milnor Bocksteins, Qn. The k-invariants of BP are given by all of the higher order oper ations which come from the Milnor Qn 's. Philosophically, using BP instead of mod (p) ho mology should have the effect of changing this higher order information from the Qn 's into simple BP* module structure information. Thus it is reasonable to expect BP* module struc tures which contain information which is difficult to gain access to from a standard homol ogy point of view. On the other hand, BP can be expected to be a useful tool for only cer tain types of problems: those where there is higher order Qn information and it has been

21

BROWN-PETERSON HOMOLOGY

turned into accessible BP* module structure information . The next section on stable homo topy gives an example of the success of this principle . However, the process of trading inac cessible higher order information for inductive formulas does ilot make life easy. A useful way of viewing Quillen's splitting is as [Q1]

(3.19) We end this section with a few examples of formulas taken from Giambalvo's paper (G). There are more terms and formulas in his paper. p = 2.

[2](x)= 2 x- v1x2 + 2v�x3- (1v2 + 8v:)x4 + (30v1v2

+

26v1)x5

-(111v�v2 + 84vDx6 + (502v:v2 + 300v� + 112v�)x7 -(I27v3 + 960v1v� + 2299v1v2 + 1140vDx8 + (166v1v3 + 5414v�v� + 9958v�v2 + 4334v�)x9 + ..., TlR(v1)= v 1 + 2tl' c(t1)=-t1,

TlR(v2)= v2 + 2t2- 5v1t�- 4t:- 3v�t1,

c(t2)=-t2 - v1t�

1/!(t1)= t1 ® I + 1 ® t1 p

-

r

:

.

,

= 3. (3]{x)= 3x- 8v1x3 + 12v�x5 - 840v�x1 -(6560v2 - 9000v1)x9 1 +(216504v1v2 -88992vDx 1 -(5360208v�v2- 658776v�)x13 + (119105576v�v2 + 1199088vi)x15 + ..., TlR(v1) = v1 + 3t1' 17R(v2)= v2 - 4v�t1- 18v�t�- 35v1t:- 21t1 + 3 t3, c(t1)=-t1, c(t2)=-t2 + t1, 1/J(t I) = t 1 ® 1 +1 ® t I , l/l(t2)= t2 ® 1 + 1 ® t2-v1t� ®t1-v1t1 ® t� +t1 ® t:.

Part II. A Sampler

4. Cooperations and stable homotopy. We begin our sampling of BP with a closer look at BP*BP with an eye to applications. There is an obvious interesting object:

ExtZj.

(4.1)

•

sp(BP.,A) s:H * * (A),

where A is a BP.BP comodule. We wi11 concentrate mostly on sim ple cases of A = BP. /1

(4.2)

where I is an invariant ideal, i.e., an ideal which is also a subcomodule of BP•· Since the coproduct (1.40) (4.3) is

a left module map, I is invariant if and only if IBP.BP � BP . BPI.

(4.4)

If we want to verify I is an invariant ideal, by symmetry it is enough to check that (4. 5 )

Thus it would be useful to know 71R(vn) (since 71R is an algebra homomorphism). We give a formula which we shall use many times. THEOREM 4.6 (RAVENEL (R1) ).

L

Ft

i�O i>O

1 177R (vi) P =

PROOF. From 3.15 , pmn

=

F v1tl L >

i O j�O

'I;O
and 3 .ll(i),

23

mo d(p ) .

Apply

71R•

o

24

W.

STEPHEN WILSON

Add over al n, substituting 3. 1 5 in the left-hand side,

Rewrite this as

Apply e xp ,

Reducing mod (p) w e are through by 3.16. o This is a useful formula for e xtracting information about 11R(- ) . LEMMA 4.7. The ideal In

=

(p, v1 ,

. . •

, vn_1)

C BP* is invariant.

o

PROOF. We need only show that 11R(vi) EinBP*BP, 0 <.j < n. Inductively it is enough to show 11R(vn-t) EinBP*BP. We will show a little more . LEMMA 4.8.

11R(vn) = vn mod In.

o

PRooF. Inductively we can assume In is invariant and we have InBP*BP = BP*BPin n so working modulo In makes sense. Use 4.6 in degree 2(p - I) modulo In; 4.8 follows. o This shows 4.7 inductively. o Using Ravenel's theorem on Lemmas 4.7-4.8 is overpowering. THEOREM 4.9 (LANDWEBER [L2) , MORAVA [Mo1) ) . The invariant prime ideals are

In, 0 <.n

<,oo.

o

SKETCH OF THE PROOF. We know In is invariant and prime . Assume I is an invariant

prime ideal. Inductively assume In

I -In. If we could show

C I; we show that if In :/:I, then In+ 1

C I. Let y

E

(4 .10) then we would have v! EI and since I is prime , vn as well; thus In+t C I. A proof of (4. 1 0) is not difficult at this point; however, it does require some serious bookkeeping to 2 know how to choose t1 = t 1 t · • · given y in terms of v's. We refer the reader to the ori ginal proofs or the more recent [JW2 ] . o Having meandered into operations only a little way, we are already in a position to state a rather nice internal structure theorem on BP*X.

{ �

BROWN-PETERSON HOMOLOGY

25

LANDWEBER DECOMPOSITION THEOREM 4. 1 1 (LANDWEBER [L3 ) ) . For X a finite complex, there exists a finite sequence of subcomodules over BP* BP, 0 = M0 C M1 C · · · Mn = BP* X, with

M;+ t fM; � BP*/In ; •

i � 0, oo > n; � 0.

C

o

The problem of 4. 1 arises naturally from results like 4.7-4.9. Let I be an invariant ideal ; then

(4 . 1 2 ) Let

a

E BP* /I determine such a map. Then we have the commuting diagram BP*

--

BP�P ®8p. BP*

(4. 1 3)

where , since the comodule maps are BP* module maps we have

(4. 1 4)

1

-- 1

®

1

1

a -- a ® 1 = 1 ® a so

(4. 1 5) determines H 0BP*/I.

THEOREM 4. 1 6 (LANDWEBER [L2 } ). S KETCH OF THE PROOF. By 4.8 we have FP [vn ] C H 0 BP*!In . For other a E BP* - In , by (4. 1 0) , 11R(a) ¥ 1L (a) mod In . o This makes H kBP*/In into an FP [vn J module . Since Fp [vn ] is a P.I.D. we know that H kBP* /ln is a direct sum of free copies of Fp [vn ] and cyclic modules, Fp [vn ] /(v! )- This 1 develops a craving for the next step, H BP*/ln , which , for reasons that will be clear in a moment, seems accessible. Be fore we move on to the higher Ext groups we should have some method of computation [M] . Le t (4. 1 7) n

c o p ie s

26

W. STEPHEN WILSON

be the cobar resolution with

(4. 1 8)

n d('y 1 1 · · · 1'Yn l a) = l i'Y 1 I · · · I'Yn l a + L 1:)- l i'Y1 1 · · · h;I'Yi' l · · · l'Yn l a i= 1

where

(4.1 9)

1/I('Y) = L 'Y ' ®

'Y

"

and

1/l(a) = L a ' ® a " .

The homology is H**A. For A =

(4 .20)

BP.JI. we have n°A = BP./1. If a EA, then 1/J(a) = 11L(a) , so d(a) = l la - a l l = 11R (a) - 11L (a) .

Just as in (4. 1 5), we find H0BP.II is the kernel of

(4.21 ) Observe that the resolution has n o elements i n internal degree q =I= 0 mod 2(p This property is called "sparseness", and is quite nice [TZ2 ] The short exact sequence •

-

1).

(4.22) gives rise to a "Bockstein" long exact sequence :

(4.23)

This begins

(4.24)

vn - 1 H 1 BP* /'ln_ -+ H 1 BP* /I H 1 BP.1In _ 1 --+ 1 n

)

We see that 6 n _ 1 (v ! ) =I= 0, k > 0, and gives al elements in H1BP. 1In _ 1 which are killed by vn_1 • The structure theorem for modules over the P.I.D. F (v - l gives hope for finding

P n t

BROWN-PETERSON HOMOLOGY

27

the entire group. All we must do is compute the FP [vn l ] free part and find module gen erators, X;, such that v��fX; = o n l (v� ) - The free part will be discussed later. Finding generators X; is j ust a matter of dividing o n I (v� ) by vn l as much as possible. We give a simple example . To compute o n _ 1 , we need to know 11R (vn) modulo /n l , as

(4.25) LEMMA 4.26 (MILLER-WILSON [MW] , RAVENEL'S PROOF [R1 ] ).

11R (vn ) == vn + vn _ 1t(

1

- v�_ 1 t 1 mod /n- t •

o

This is a good example of skimming off information from the inductive formulas. This formula contains a tremendous amount of information and can be pushed a long way for results ; see [MW] . PROOF. We use Ravenel's equation 4.6. In the degree of 11R (vn), the left-hand side is, modulo /n 1 ' 11R (vn _ 1 ) +F 11R (vn ) +F t1 11R (vn - l )P, which is simply 11R (vn ) + t 1 11R (vn _ 1 )P , and by 4.8, this is 11R (vn) + v :_ 1t1 • -1 nThe right-hand side of 4.6 is vn _ 1 + F vn _ 1t r +F vn which is vn + vn_ 1 t f l . o s Now we can present our example of some dividing by vn _ 1 • Write k = ap , a =F 0 (p).

We know

(4.27)

1

Clearly v:�1 divides this, so the element (4 .28) is nonzero and defines for us an injection of FP [vn _ 1 l /(v:� 1 ) . We have easily come up 1 with a large number of elements in H BP.IIn - t · These elements, for n = 2 , were found in· dependently by several (6) people , but first by Zahler [Zh] . For awhile it was believed this was all of the vn - J torsion . Haynes Miller gave an example of a new element for n = 2, and eventually the entire structure fell [MW] . When the above a = 1 , there are no more ele ments, but when a =/= I , one can divide (a homologous cycle) by vn_ 1 more times. A favor 1 ite pictorial representation for the vn_ 1 torsion in H BP .1In_ 1 is

W. STEPHEN WILSON

28

(4.29)

top

e elements o n_ 1 (v�)

line gives all of th where multiplication by vn _ 1 is vertical and the killed by Vn_ • 1 Let us do a little more computing. Let p be odd. For H1 BP* we have (4.30) Then (4.3 1 )

1 1 is divisible by p 8 , and 5 0 (vfP8)/P 8 reduces t o avfP 8 - t1 , a =F 0 (p), in H BP.f(p). The v 1 1 free part of H BP.f(p) is generated by t 1 (and there are no Z(p) summands of H 1 BP.). Therefore , (4.3 2 ) This looks like the image of J at an odd prime. Seeing this, we know it is time to leave our discussion of rather sophisticated internal theorems about the cohomology of BP*BP and start to look for applications. We need the Adams-Novikov spectral sequence . by the stable cofibration: Define

0

-

S(p ) -- BP -- BP.

(4.3 3)

We form a sequence of stable cofibration triangles with the top maps all changing degree:

(4.34)

BP

BP A BP

BP I\2BP

BROWN-PETERSON HOMOLOGY

29

Smash this with X and apply 1f *(- ) . The resulting spectral sequence has (4.35) * For H BP* � 1r*S& )• p > 2, the calculation of H1BP* really does correspond to the image of J [N] . Other parts of the previous algebraic material correspond to geometric situ ations. For the first example, Larry Smith generalized the elements of order p in the image of J. He considered spaces V(n) with mod (p) cohomology E(Q0, Q l ' . . . , Qn), or equiva lently [Sm 1 ]

(4.36) This is the situation discussed at the end of the last section as most likely to be accessible by BP techniques. Unfortunately, the V(n) spaces are only known to exist for very small values of n , but in these cases they have produced a great deal of homotopy information. Letting S0 = V(- 1 ), the V(n) are constructed inductively as cofibrations, (4.37)

� 2 (pn - l ) V(n

I ) � V(n

I) � V(n) ,

which realize the exact sequences (4.38)

0 � BP*/ln

un

--

BP* /ln � BP*/In + l � 0.

The space V(O) exists for all p, V( I ) exists for p > 2 [A3 ] , V(2) for p > 3 [Sm2 , To1 ] , and V(3) for p > 5 [To 1 ] • The elements of order p in the image of J are constructed by the composition (4.3 9 )

k � 2 (p - l )ks o � � 2 (p - t )k V(O ) � V(O) � s t

obtained by iterating the map in (4.3 7). The map to V(O) is represented by (4.40) in the Adams-Novikov spectral sequence. The elements of order p in the image of J are represented by (4.4 I ) sition

Larry Smith constructed and detected the elements f3k [ Sm2] defined by the comp o-

(4.42)

pk s 2 (p 2 - t )k � � 2 (p 2 - t }k V( l ) -V(l ) � � 2 (p - I ) + I V(O) � s 2(p - t } + 2.

The map to V( l ) is represented by (4.43) The nontriviality of this composition is determined by the n ontriviality of (4.44)

W.

30

STEPHEN WILSON

We have seen that 6 1 (v�) =I= 0. We then have an exact sequence (4.45)

6

-+ H 1BP* -+ H 1BP.f(p) � H2BP* -+

where we know the first two groups and can easily verify that 6 0 6 1 (vn =I= 0. See [JMWZ] or these notes after (4.3 1 ). The element 'Yk is defined by the composition (4.46) The map to V(2) is represented by (4.47) It is enough to show that (4.48) There is a result involved in making the leap from algebra to geometry . The place to find it is in [JMWZ] . We already know that

most

accessible

(4.49) We consider the sequence (4.50) We know the first two groups and so we can verify that (4.5 1) Our next exact sequence is (4.52) To prove that (4.53) it turns out to be necessary to compute H2BP. , but fortunately not H2BP.f(p). There is a pictorial representation for H 2BP* corresponding to that for H 1 BP*/In given already. We picture H 1BP* I(P) more explicitly :

BROWN-PETERSON HOMOLOGY top

tow

31

is Ei 1 (v�)

(4.54)

free summand on

t1

'-

v1

multiplication is vertic::tl

H 2BP* is all p torsion. Using (4.3 1 ) to compute (4.45), we see that the image of H 1 BP* in H 1BP .f(p) is the free tower on t 1 • Thus all of the u torsion part of H 1BP.f(p) maps to 1 H 2BP* and gives the elements of order p in H 2BP * ' All that remains is to divide these ele ments by p as much as possible. The exact divisibility can be found in [MRW] . The v1 tor sion part of the picture for H 1 BP.f(p) is shifted to H2BP•. The main elements are Smith's {jk = o 0 o 1 (vf). From {jk with k = ap 8, s > 0, there are hairy stalactites hanging down. The hair represents the division by powers of p. Before proceeding to the description of the approach to the computation of H2BP* this seems like a good place to sketch a simple proof of the fact that 'Yt =F 0 [TZ1 , OT, A4 ] . This element generated a great deal of interest when the original papers of Oka-Toda and Thomas-Zahler came to opposite conclusions. The proof given here has evolved over the years with all of the advantages of hindsight with contributions from many people. Nothing is used that was not available to the combined forces of bordism and stable homotopy ex perts at the time of the original proofs . The bordism experts had known for some time that

(4.5 5)

0 =F 'Y l E 1TI'Y t l + 1 V(O)

in complex bordism (BP) filtration two, i.e.

(4.56)

W.

32

STEPHEN WILSON

The only remaining question was whether this lived on the top or bottom sphere of V(O) in the exact sequenc (4.57)

Both of the end groups are in known stable stems isomorphic to Z/(p). The only question is if -y 1 is hit by the left-hand Z/(p ) . The stable homotopy experts knew that this group was generated by 13f [To2 ] . The filtration is thus greater than two so it cannot hit -y1. The complexity of the computation of H 2BP* necessitates some new techniques. We begin with some exact sequences which define the new BP.BP comodules:

(4.58) 0 -+ BP.f(p ""' , v�) -+ v:21BP.f(p ""' , v�) -+ BP.f(p""', v� , v;) -+ 0

Apply H *(-) : H *BP*

(4.5 9)

/

\

v �)

I

\

I

\*

H v i 1BP.!(p""' , v�)

This gives a spect ral sequence (4.60)

called the chromatic spectral sequence [MRW] . components. Since H *p -I BP* � Q, we have

It

breaks H *BP* up into its

vn -periodic

..

0

(4.6 1 )

.. ..

..

' H�21BP..f(p "" , u;'") ..

33

BROWN PETERSON HOMOLOGY

To get H2BP* we need H1 v}1 BP* /(p"") and H0 v:; 1 BP* /(p"", v;"). This is already a great improvement (despite the complexity of the modules) because the lower the cohorriological degree, the more accessible things are . In particular,

H0M C M.

(4.62)

The first differential kills all of H 1 v -1 BP*f(p "" ), so 1 H0 v:; 1 BP*/ (p "", v�). We have an exact sequence

H2 BP*

comes only from

(4.63) Apply H

*(-) to

get a Bockstein exact couple

H*v } 1 BP*/(p"") � H*v } 1 BP*/(p"") (4.64)

Likewise , we have

\

I

(4.65)

*(-)

Applying H

we get two exact couples, the second beginning with

(4.66) In general we start with the

vn torsion

free part,

(4.67) and have

n

Bockstein spectral sequences to get to

(4.68)

*

The computability of (4.67) is a key motivating factor in this approach to H BP*. This computability is part of the Morava structure theorem for complex cobordism. We give the Miller-Ravenel approach. THEOREM 4.69 (RAVENEL [� ) }.

Let K(n)*

=

FP [vn ,

v,;1 ) .

W.

34

STEPHEN WILSON

PROOF. It is elementary that

We go to Ravenel's formula 4.6

L

F

f;flR (vi)P

i

=

F

L v; tf

i

mod (p). i> O i"> O n i i This reduces to �f-. o t; flR (vn )P = �� o vn tf . By induction on degree we have t;flR (vn)P n = vn tf , but by 4.8, flR (vn ) = vn , and so i� O i> O

_

THEOREM 4.70 (MILLER-RAVENEL [MR) ) * H v;; 1BP.1In � v;;1 H BP*/ln . o

*

.

ExtK (n ) . K (n ) (K(n)* , K(n)*)

�

Proving this is beyond the scope of these lecture notes. Morava's approach (the original) is dual to this [Mo1 , Mo 2 ] . The "dual" of K(n)*K(n) is the "group-ring" for the p-adic Lie group (known as the Morava stabilizers) of endomor phisms of the height n formal group law over the closure of F . The continuous cohomol P ogy of this Lie group determines the cohomology in 4.70. A good explanation of the details of this duality is in Ravenel [ � ] . Both approaches give H 1 v;; 1BP.IIn easily . Another exciting motivating factor for the study made in [MRW) is the following im· portant finiteness property which gives a parabolic vanishing curve (with spikes) for the chro matic spectral sequence. THEOREM 4. 7 1 (MORAVA

[Mod ).

If p

-

1 does not divide n, then

2 H 'v;;1 BP.lin = 0 for t > n •

o

We are leaving the computations of cohomology groups over BP*BP. The computation 2 of H 1 BP.lin and H BP* demonstrates a good grasp of the operations for BP. Furthermore , these computations have led to many concrete applications in stable homotopy. S. Associated homology theories. There are many generalized homology theories as sociated with BP. Each family of theories takes us in a different direction, but the theories themselves have a common origin. For this origin we must revert to geometry. Recall that MU *X is given by equivalence classes of maps (1 .6) :

(5 . 1 ) I t i s Sullivan's idea [Su) t o kil manifolds selectively to form new generalized (bordism)

homology theories. Baas theories. Let

(5.2)

[Ba]

developed a good bookkeeping system for dealing with these

BROWN PETERSON HOMOLOGY The b ordism theory

MU(x2 n )* X fits into an exact sequence

j

(5.3)

with MU(x2n ) * with boundary

::

MU* f(x2n ).

A "manifold with singularity" i n MU(x2 n )* X i s a U-manifold

V,

(5 .4) and a map V (5 .5)

35

X such

that on

a V the map factors

through th e projection

M x P - P - X.

From Sullivan's point of view this is the same as allowing the cone on M to be a manifold with boundary M. This can be generalized to define "bordism" for pairs of spaces. The veri fication that we have a homology theory and the exact sequence (5 .3) all come from the geometry [Ba] . Inductively we can define (5 .6) with coefficients (5 .7) and all possible long exact sequences. The multiplicativity of these theories is dealt with in [SY, Wu1 , Mo3 ] . Choose an infinite sequence of x2 i 's which excludes all of the x2 n - ' all n , fixed (p 1 ) prime p. In the limit, after localizing at p, we have constructed BP* X! Kill off a few more generators and we can construct theories with coefficient rings

(5 . 8)

36

W. STEPHEN WILSON

The last two are due to Morava. These theories come with exact sequences

I

BP *X

I

(5 .9)

P(n + l )*X

H*(X; Z/(p)) These long exact sequences al give Bockstein spectral sequences. We have seen this before. The first direction we pursue using these associated theories has not really moved out

of the internal theorem stage. However, there are many beautiful internal results and much has been written about it. See [CS] and [JW 1 ] for the basics. We define hom dim8p

(5 . 1 0)

•

BP*X

to be the minimum length of a free BP* resolution for BP.x: (5. 1 1 )

The study o f this number (for MU. X) was initiated by Conner and Smith in [CS] , where =

0, 1 and 2 cases are investigated thoroughly. Unfortunately it turned out to be necessary to localize at a prime and go to BP in order to continue the study. This number the n

measures the complexity of the

BP*

module structure and so. is sometimes referred to as the

"ugliness number". The main characterization is in terms of the associated theories BP.x.

37

BROWN PETERSON HOMOLOGY Consider the sequence

I

(5 . 1 2)

THEOREM 5 . 1 3 (JOHNSON-WILSON [JW1 ] ). hom dimB p . BP* X

are all surjective and p0 , p1 ,

• • •

, Pn-1 all fail to be surjective.

o

=n

�

Pn • Pn + l • . • .

Other related results are THEOREM 5 . 1 4 (JOHNSON WILSON [JW1 ] ). hom dimBr BP. X > n. o

If X E BP* X is vn torsion then

Although any expert tends to feel it should be obvious, it is only recently that a proof has been found for THEOREM 5 . 1 5 (JOHNSON-YOSIMURA [JY] , SEE ALSO [L5 ] ).

torsion then it is vn _ 1 torsion.

o

IfX E BP* X is Vn

The proof is not obvious. Along this line is a possibility that any n dimensional class in BP.x (for X a space) is not vn torsion. David Johnson first raised this question. There is some minor evidence for it as a conjecture . It is a very strong statement about the possible unstable BP* module structures. In this direction the most interest has been centered on the theories (5 . 1 6)

1

where u;; means localization with respect to { u: } , i.e., inverting vn , first studied in [JW. J and sin ce studied in [L4 , R3 , JY] . The n = 1 case of (5 . 1 6) is the BP version of the Conner Floyd theore m [CF 2 ] showing how complex cobordism determines complex K-theory : (5. 1 7)

KU *X :;: KU * ®M U * MU *X.

The n > 1 cases of (5 . 1 6) generalize the Conner-Floyd theorem in one direction. These theories are vn -periodic and one can see similarities between this type of thing and the chro matic spectral sequence. In particular we have the recurring theme of using one vn at a time and building up slowly. We will see a lot more of this in the next section.

38

W. STEPHEN WILSON

6 . .Morava's litde structure theorem and the Conner-Floyd conjecture. We take an other direction using associated homology theories. We have already mentioned Morava's structure theorem in the setting of computing Ext groups. The Morava structure theorem is a vast internal sequence of theorems by Morava about complex cobordism and related alge bra. It is an internal theorem with long range consequences and applications in many direc tions. Here we will ignore Morava's work with operations and the relationship with stable homotopy and give Morava's little structure theorem and tie it in with a new direction in our study to give .a proof of the Conner-Floyd conjecture. Because this calls for so much machinery it usually cannot be presented to an audience unless they have been properly warmed up to BP. However, it is one of the most beautiful and complex applications of the theory so it is a central theme in these lectures. Published Morava references tend not to exist, but many of his results (in preprint form) date back to 1 9 7 1 or so . For Morava today, see [ Mo d and [Mo1 ] , hoth presently still preprints.

THEOREM 6 . 1 (MORAVA, SEE FP [u; l , vn , un + • . . ] . o

l

(JW2 ) }. B(n} *X := v; 1 P(n) * X is free over B(n) * �

.

Plausibility argument. P(n ) . X is approximately a comodule over BP.BP/1n . There are no nontrivial invariant ideals in B(n)* because, as in (4. 1 0), v! would be in such an ideal but it is a unit. So, it is plausible that B(n ) . X is free . o THEOREM 6.2 (MORAVA, SEE

(JW2 ) } .

Here is another generalization of the Conner-Floyd theorem (5 . 1 7) (the n = 1 case is just the mod (p ) version of (5. 1 6)). COROLLARY 6.3 (MORAVA, SEE o == K(n).x ®Fp P(n + 1 ). .

Wurgler

(JW2 ) }.

There is an unnatural isomorphism

B(n) * X

[Wu2 ] has made this reversing process precise.

MORAVA'S LITTLE (NO OPERATIONS) STRUCTURE THEOREM 6 .4. The Vn torsion free part of P(n).X is determined by K(n ) . X {63). The Bockstein long exact sequence (5.9) re lating P(n).X and P(n

+

I).X gives the vn torsion. For BP.X, the diagram is

BP,\ 1

•

39

BROWN-PETERSON HOMOLOGY

BP,X

_ _

;. p - • BP,X � H,(X ; Q)

®z (P)

P( l )*X

�P(I )*X - --4 v} 1P(1 )*X � K(I )*X ®F P(2)* . I P(2).x 6

\

6

I

.

P(2)*X - - -4 v2 1 P(2) *X � K(2) * X ®F P(3) *

P(3)* X I I I

P(n).x 6

�

P(n)*X ---4 v;;1P(n) . X � K(n).X ® P(n + 1 )*

I

P(n + I ) .x

Although there are no operations see

m

0

the statement, they are in the proof. Again we

vn -periodicity in this structure theorem.

Up to this point the result is purely internal. An attempt to use this to compute

BP.x for some

space

X would convince

most people that it really is only an internal result.

However, we will use it to prove the Conner-Floyd conjecture which we now digress to de scribe .

W.

40 Let

G be

STEPHEN WILSON

p

a finite abelian group of order p ",

odd. Defme

SF(G) C MSO *

(6.5)

to be the ideal given by classes of manifolds with a differential orientation preserving action of G with no stationary points.

MSO * X is defined like MU* X except

using SO structures

on the manifolds. There is a canonical element -r:

(6.6)

s1

x ·

·.

x

n co p ie s

'Y

s1

-+

x

BZ/(p) x

E MSOn (BZ/(p)

··· x

BZ/ (p) ,

n c o p ie s x

· · ·

n co p i e s

BZ/(p )) .

The annihilator ideal is defined as: Ann

(6.7)

'Y

=

{x

E MSO.I x-y =

THEOREM 6.8 (CONNER-FLOYD [CF1 ] ).

SF(G) C

0} . o

Ann('Y).

i of SF(G). Let M 2 (p - t )

Thus, Ann('Y) is of interest for the study

be Milnor basis

elements in Mso • .

THEOREM 6.9 (CONNER-FLOYD [ CF . J ).

2 (p, M 2 (P - l ) , M 2

) C

Ann(-y).

o

THEOREM 6. 1 0 (RAVENEL·WILSON [RW2 ] ). THE CONNER-FLOYD CONJECTURE [CF 1 ] .

(p,

M 2 ( p - l ) , . . . , M 2 (P

n-1

- l )) =

Ann('Y).

o

Floyd and tomDieck eventually by-passed the Conner-Floyd need for this with

G2 $ · · · $ Gk , G; cyclic, then ( k t (p, M 2 (p - l ), . . . , M 2 p - _t > ) . o

THEOREM 6 . 1 1 (TOMDIECK [tD] ) .

SF(G)

=

Floyd [F] first proved the

If G

G = Z/ (p )

x

=

···

G1 x

$

Z/(p ) case .

The group

MSO.(BZ/(p)

(6. 1 2)

X • • • X BZ/(p))

is the group of oriented manifolds with free actions of Z/(p) x · · · x cobordism. At odd primes the relationship between MSO* X(

to that for MU* X(

by the map

MU

•.

p) ·

P)

Z/(p) on

them, up to

and BP* X is completely analogous

In particular, we have MU*X ®Mu.

MSO *(p ) !: �SO * X(P )

induced

(1 .3) (cf. [St, p. 1 80 ] ), p odd. Thus the elements [M2

] come from

The conjecture is completely a p-primary problem, so it is the same as:

41

BROWN-PETERSON HOMOLOGY COROLLARY 6 . 1 3 (RW2 ) - Ann('Y) = In

in

BP.(BZ/(p) X • • • X BZ/(p)). n c op ies

0

There is a rather obvious approach to this problem. Just compute

BP.(BZ/(p) X

(6. 1 4)

•••

X BZ/(p))

and look a t the answer. This has the advantage of giving the cobordism groups o f Z/(p) x · · x Z/(p) free actions on oriented manifolds. However, unlike the operations discussed in the last section, the computability of BP* X is in very bad shape. The only spaces that could be dealt with reasonably until now were torsion free spaces (the Atiyah-Hirzebruch spectral sequence collapses), spaces with few cells (such as V(n)) and a few spaces like BZ/(p). In particular, what was needed was an infinite sequence of spaces with known BP homology and increasingly complex BP* module structure. Since these lectures were given , David Johnson and the au thor have succeeded in computing (6. 1 4). Examples of this type are necessary in order to build the sort of confidence in the computability of BP.(-) that standard homology presently enjoys. The labor (and the 1 6 year wait) that the reader will see us go through to understand something of one element in (6. 1 4) and prove the Conner-Floyd conjecture illustrates the poor state of computability for BP*( ). However, the understanding of this element is crucial to the computation of the entire group (6. 1 4). This new computation adds even more to the complex chain of results which ends with these concrete geometric applications. To prove 6. 1 3 we take the map n (6. 1 5) x BZ/(P) - Kn = K(Zf(p) , n) ·

which takes

'Y

to 'n .

CoROLLARY 6 . 1 6 (RW2 ) . Ann(tn ) = In ,

tn

E BPn Kn ·

o

All of the above follow from THEOREM 6. 1 7 (RW2 ) . Ann(tn) = In , tn E

v;; 1 BPn Kn · o

To prove 6. 1 3 and 6 . 1 6 from 6.1 7 we just apply the maps and 6.9 to get (6. 1 8)

In

C

Ann('Y)

C

Ann('n )B P

C

Ann(tn )

v; 1 BP

= In .

To prove 6 . 1 7 we compute (6. 1 9) in its entirety . To do this we use the Morava structure theorem. We start with the Morava K-theories of Kn (developed more later)

(6.20)

K(n ) * Kn � a K(n) * free module on p - 1 even generators.

W.

42 To compute , we invert

vn

STEPHEN WILSON

in the Morava Structure Theorem 6 .4 using reduced theories in

each place :

v;;

1

.

BP*Kn

(6.2 1 )

After this localization we have reduced the computation process to a finite number of Bockstein exact couples. By (6.20) we have

(6.22)

v;; 1 P(n) * Kn =:! p - 1

1 copies of v;; BP.Iln on even degree generators.

BROWN-PETERSON HOMOLOGY We now compute v; 1 P(n - l)*Kn · We see from (6.20) and sion. We use

43

(6.2 1 ) that this is al vn_ 1 tor

(6_23)

From (6.23) the vn_ 1 torsion must be in odd degrees because the degree of 6 is odd, the degree of vn_ 1 is even, and v;; 1 P(n)*Kn is in even degrees. Thus p is zero and (6.23) is a

short exact sequence which is p

- 1 copies of

(6 . 2 4)

So (6.25)

on odd degree generators.

Repeat this type of computation until we have

(6 .26) Note (4.63)-(4.68) the familiar need to locate

'n .

BP*

modules! To finish the Conner-Floyd conjecture we

This is just a homology computation.

The problem of computability is a very serious one . Consider the (time and location dependent) functor (6.27)

topologists .. topological spaces

which assigns to each topologist his favorite topological space. If an algebraic theory E . (-)

X must be able to compute E.(F(X)). After all, if he wants to prove F(X), E.(-) does no good if he cannot get his hands on the algebraic in variant E.(F(X)). By this standard it is clear that H.(-) is fairly computable , but it is not clear why, since there is no workable algorithm for computing H.(F(X)). As a substitute for an algorithm, Y has had to rely on the observation that X has computed H* (F(X)) for all X older than Y. This supplies a certain faith that Y too can compute H.(F(Y)) , even if is to be useful, then theorems about

he must do it in an "ad hoc" fashion and be prepared to spend two years at it. With H * (-)

he has the decades of development of homological algebra, the Steenrod algebra, homology

operations, etc., to help him out. In general, although there are millions (>> oo) of homology

44

W. STEPHEN WILSON

theories, very few are computationally useful. After H.. (-) we find that people have been quite successful with K-theory. Now , with (6.28)

K(n) .. K(Z/(p), j)

and

BP.. (BZ/(p ) x · · · x BZ/(p ))

computed there is some hope for K(n) .. (-) and BP.. (-). It may be that BPi-) never be comes an easily computable theory . We see by the above example that K(n) .. (-) is some

times an acceptable substitute. It appears to be quite computable , due mainly to the

Kunneth isomorphism (6.29)

which follows because K(n) * is a graded field. We discuss (6.28) more later.

7. Hopf rings, the bar spectral sequence and K(n) .. K.. . The techniques for computing

and describing K(n) * Ki ( Ki = K(Z/ (p ) , j )) are general and have several BP related applica tions (like Sullivan's theory of manifolds with singularities). We have already used the Morava structure theorem for the proof of the Conner-Floyd conjecture. In addition, we need all o f the material i n this section t o compute the necessary K(j) * Kn for the completion of the proof. After this discussion we can bring up some other examples of the technique and this will eventually lead us to unstable cohomology operations. Let

(7. 1 ) b e a n n-spectrum, i.e . (7.2) This represents a generalized cohomology theory

(7.3) We want to study

(7.4) where E..(-) is a multiplicative homology theory. To apply our tools we need a Kiinneth isomorphism for the spaces

{ik .

This condition seldom holds in general, but always holds

for

(7.5)

E..,(-) = K(n) ..(-)

or

H.. (- ; Z/(p)).

It also usually holds for

Q*

(7.6)

or

BP* .

These techniques can be used to compute things like

(7.7)

K(n) *K* ' E ..Mu..

.

E*BP*,

H*BP< n> .. .

We feel that even more non-BP applications could be made.

and

H*K* ·

45

BROWN-PETERSON HOMOLOGY

We have that G kX is an abelian group. Thus gk must be a homotopy commutative H-space (not surprising since we already know it is an infmite loop space), or, in other words, an abelian group object in the homotopy category . By our Kiinneth isomorphism, E.(-) takes gk to a coalgebra ; it also takes products of gk to products in the category of coalgebras (tensor product). Thus E . (-) takes the abelian group object gk to an abelian group object in the category of coalgebras. This is just to say that E*gk is a (bi-) commu tative Hopf algebra with conjugation. This "abelian group" structure , or the Hopf algebra multiplication, just comes from the product (7 .8)

after applying E * (-) : (7.9) Considering everything at once , G *X is a graded abelian group so, as above, graded abelian group object in the homotopy category . Likewise ,

g.

is a

(7 .1 0)

is a graded abelian group object in the category of E* coalgebras. Of course, when G a ring spectrum, G *X is more than just a graded group ; it is a graded ring. Then the graded abelian group object g. in the homotopy category becomes a graded ring object. The multiplication

is

(7. 1 1 }

has a corresponding multiplication in

g.:

(7. 1 2)

Applying E. (-) we have

(7. 1 3) turning E *g* into a graded ring object in the category of coalgebras. This object "should" be called a "coalgebraic ring" but instead has taken on names such as "Hopf bialgebra" and the one we use , "Hopf ring". See [RW 1 ] A ring must have a distributive law, and ours is: •

(7. 14)

x o (y * z ) = L ± (x ' o y) * (x " o z) where 1/l{x) = E x ' ® x ".

46

W. STEPHEN WILSON

Because of the importance of the relationship between the two products and the coproduct * we will track it down from the distributive law in G X. We have that

l XX G n x G kX 1I X X diag

(7 . 1 5)

G nx

x

x

G kX

is the distributive law in

G *X.

G nx

(7.16)

I

I

x

x

switch

I

x

G nx

x

G kX G kX

o

X

o

In terms of classifying spaces it becomes

l.O. XI XI Gn x Gn x {jk x {jk

Apply E.(-) to obtain (7. 14). Because of the two products it is easy to construct many elements starting with just a few ; this helps to describe answers in terms of Hopf rings. All of our examples will demon strate this property in a strong way. We will also demonstrate some of the other benefits which allow the Hopf ring structure to be used seriously in proofs as well. For even deeper applications we will put the Hopf ring structure into the bar spectral sequence. To do this we review the bar construction . Let a n be the geometric n-simplex and {j� the zero component of {jk . Then

(7.1 7)

g� + l � BQk � 11 a n n ;;>O

x

Qk � Qk /n COpieS •

where - indicates that there are identifications made [Mg] which we will not give explicitly in these lectures. B{j k is filtered by

(7. 1 8)

B/i1c

�

11 a n

s;;> n ;;>O

x

Qk

x ··· x

�

n

{ik /- C B{jk .

copies

Apply E.(-) to this filtered space to get the bar spectral sequence (RS]

(7.1 9) we have

(7.20)

s copies

.

Since

BROWN-PETERSON HOMOLOGY

47

and (7 2 1 ) .

This is a spectral sequence of Hopf algebras. To put the additional structure of the plication in the bar spectral sequence we look at

u

o

multi

u

(7.22)

THEOREM 7 .23 (THOMASON-WILSON (TW) ) . The

0

product factors as

n

n

and the map

'i:/G -k

I

1\ . . 1\ -Gk X Gn - �.I'G - k+ n .

s cop ies

is described inductively as (g 1 ,

• • •

,

gs)

o

g

=

(g 1

o g,

1\ . . . 1\ -Gk+ n s copies

. . . , gs

o

g).

o

This automatically implies: THEOREM 7 .24 (THOMASON-WILSON [TW] ). Let E� * (E* f!k ) bar spectral sequence. Compatible with '

is a pairing

�

E.f!� + 1 be the

48

W. STEPHEN WILSON

with dr(x)

o

y = dr(x

o

y). For r = 1 this pairing is given by

(g l I · · · I K8) o g = 1: ± (g 1

o

g ' lg2

o

g"l · · · lg8

o

g (s ))

" where g-+ 1:g ' ® g ® · · · ® g(s ) is the iterated reduced coproduct.

o

This result is easy to prove, but tremendously powerful. It allows one to identify ele· ments in terms of o products, compute differentials inductively , and solve extension problems using Hopf ring properties. This will all be demonstrated . The first example of the above pairing was given in [RW2 ] with q * = K* . It is unnecessary to know how to prove this re sult in order to use it for computational purposes. We wish to inject some formal group nonsense at this point . Our formal gro ups all come from the standard map (2 .3): J.l :

(7.25)

CP""

X

CP "" - CP "" ,

which i s unstable information. This is generally stabilized immediately, with the exception * of Quillen's proof that MU X is generated over MU * by nonnegative degree elements [Q2 ] . We go even further in [RW 1 ] in extracting unstable information from formal groups. In E * Q * we have two formal groups. Later we will get unstable relations from their interaction. For now, we concentrate on just one formal group. We assume E is complex orientable, i.e. CP"" has x E and f3f just like MU in (1 .26) and (1 . 30). Thus we have a formal group law * for E and elements a ff . This formal group is the coproduct in E CP"" . Dual to this is the product in E * CP"". For E * (-) = H* (- ; Z ) we can let {J(s) = 1:;;;. o f31 s1 and describe this product as

{3(s){3(t) = {3(s + t).

(7 .26)

1i We know H* (CP .. , Z ) is a divided power algebra. If we look at the coefficient of s t we get (7.27) so the notation has managed to hide the binomial coefficients. We have much more to hide. Let

{3(s) = .L f3fs1 E E * CP .. [ [s] ] .

(7.28)

;;;.o

THEOREM 7.29 (RAVENEL-WILSON (RW1 ] ) . In E * CP"" ( (s, t] ] ,

{J(s){3(t) = {3(s +F t).

o

PROOF. Write {31{3i = 1:k ck {3k and (x 1 +p x 2 )k = 1:1, i a 1�x {x� . Then c•

�

�

�

c, x

)

c,. •x • , P, ® P;»

�

�

<x • , PN

�

<x • , ",( {11

® x \ , P1 ®

®

P.)

P;)> �

a1j .

BROWN PETERSON HOMOLOGY

P(s) P( t) = 'E;,A s ipli = 'E;,j, Jca;1s itipk P(s +p t). o The n

=

49

'E�cPk 'E;ja;1siti = 'E�cP�c(s +F t)"

=

Iterating we have : COROLLARY

7.30 (RW. J . P(s)"

=

P( (n ) p(s)) .

0

On the surface , this appears to be another nonsense formula involving totally inacces sible coefficients. However, in the important case E = BP we can extract an explicit for mula. Let (7 .3 1 )

I t i s fairly easy t o see that the other P's are decomposable. THEOREM

7.32

(RAVENEL WILSON (RW1 ] ).

In QBP*CP•

" v :n-kP(n- k ) 0= L k= l

PROOF. Recall 3 . 1 7 :

P(s)P

=

(p] (x) = r.:> 0v,xP"

(

"

mod

)

P( (p] (s) ) = P L F v, sP " n>O

= n P(v, sP " )

mod (p)

n>O

mod (p),

0

(p). By 7.3 0 , mod (p)

by iterating

7 . 2 9.

Since Po is the multiplicative identity we have P(s)P = Po and n,. >0P(v, sP") = '£,. > 0 P(v,sP") in QBP* CP mod (p). Just pick off the coefficient of sP ". o . We can now proceed to our discussion of the Morava K-theories of the mod (p) Eilenberg-Macl.e spaces, K(n ).K• . THEOREM 7.33 (RAVENEL WILSON {RW2 ) ) .

Pf;>

=

In K(n)* CP ..

0
0,

PRooF. The formal group law for BP reduces to

K(n ) to

P(v,. sP " ) = L P; v! s iP " . iOioO The coefficients of sp i +

n

give the result.

o

l,

give

so

W. STEPHEN WILSON

Recall that Ki = K(Z/(p}, j). THEOREM 7.34 [RW2 ) . The standard map K1 -- CP• induces a Hop{ algebra inclu n sion K(n ).K1 C K(n).CP-, where K(n).K1 is free over K(n) . on a1 E K(n ) 21K 1 , 0 <; i < p , which maps to ��- The standard coproduct a k -- I:ak-i ® a1 follows and

a(,!'- 1 ) = vn a ( o ) • a (i� = 0, PROOF.

o

0 <; i < n - 1.

o

Any attempt works.

THEOREM 7 .35 (GLOBAL VERSION, RAVENEL-WILSON Hop{ ring on K(n).K 1 • o

(RW2 ) }. K(n) *K*

is

the free

We have to say what a free Hopf ring is. Given a graded Hopf algebra (7.36) H.(k} a (graded) Hopf algebra with conjugation, the free Hopf ring FH.(•) is a functorially assigned Hopf ring and map a such that any map f of our Hopf algebra to a Hopf ring factors through a:

(7.37}

The Hopf ring FH. ( •) is constructed by taking all fmite

*

products of al finite o products

of elements and then using the relations obtained from Hopf algebras and Hopf rings. Since our "abelian groups" in our fancy ring are bicommutative Hopf algebras with conjugation, our "- 1 ", denoted [- 1 ] , is just conjugation. So, if we are in a Hopf ring R .( • ) with (7.38}

In the case of K(n).K1 ,

[- 1 ] a(i)

=

-a( i) • so

(7.39)

which implies, by Hopf ring madness, that (7.40)

In K(n).K• • the longest

o

product is

(7 .4 1 }

By th e distributivity law (7.42)

a(i+ 1 >

o

* x *P = (a(i) o x) P ,

BROWN PETERSON HOMOLOGY

so * a1 P

-

o

. • •

o a(n - t ) )

*P

-

a( l )

o

o

• • •

51 o a (n - 1 ) )

*P

(7 .43)

These are all o f the elemen ts in K(n ) * Kn so as advertised in (6.20), K(n ) * Kn is free over p 1 even degree generators. Furthermore , any longer o product must give zero, so since K(n)*K 1 generates everything, K{i;)*KI = 0, j > n, as in (6.20). This is all we needed to prove the Conner-Floyd conjecture . A detailed description (local version) and proof for K(n)* K* are available in [RW2 ] , and for the more difficult K(Z/(p i ), j) as well. In particular, we describe K(n) *Ki completely as a Hop f algebra. This is necessary since we use the bar spectral sequence (7.2 1 ) to make the computation . Inductively we compute Tor . Using 7.24 we can name many elements, show which are infinite cycles, and compute the differentials, all inductively. Then we use the Hop f ring structure to solve extension problems. Theorem 7.35 follows from this de tailed version . This type of calculation, which relies heavily on the Kunneth theorem, demonstrates the computability of K(n )*(-) to our satisfaction, and we recommend them to other homo topy theorists. ,

-

8. H *K* and the Steenrod algebra.

Before we introduce the second formal group law into Hopf rings we want to do a complete calculation. We will do a simple one, just the mod (p) homology of the mod (p) Eilenberg-MacLane spaces H*K* · Usually people do the p = 2 case, but we will do only the odd primes. They demonstrate the pairing 7 .24 better. Letting (8 . 1 ) when w e have H*K* w e will also have H*H, the dual o f the odd primary Steenrod algebra. Having computed H*H it is nearly obligatory to give its structure as a Hopf algebra. This is no problem since we have already done all the work for MU*MU. This computation of H*K* represents joint work with Douglas Ravenel. We came across it in the early stages of our work with Hopf rings. Since that time we have promised many pe ople that we would write it up for pedagogical purposes. We thank Doug Ravenel i for allowing us to use this in these notes. We leave the K(Z/(p ) , n) cases and Bockstein's, etc., for a potential future write-up.

W. STEPHEN WILSON

52

The beauty of this approach is that neither chains nor Steenrod operations need to be introduced. Everything is done using standard homological algebra to keep track of the Hopf ring structure inductively using the pairing in 7 .24. To describe our answer we need notation for H. K and H* CP "" . We have

1

i � 0.

(8 .2) The generators are (8 .3) The coproduct is

n ) ( = n L n ® , 1/l a i=O a -i a;

(8.4)

n

n ® 1/1 ({3n ) = iL =O l3 -i f3;·

THEOREM 8.5 (GLOBAL VERSION). H* !;* is the free Hop[ ring o n H* Ko H .K 1 , and H* CP "" C H. K2, subject to the relation that e 1 o e = {3 . o

1

=

H* (Zfp) ,

1

Denote the height p truncated polynomial algebra and the exterior algebra respectively by E(x) = Z/(p) [x ] /(x 2 ).

1P 1 (x) = Z/(p) [xJ /(x P ),

(8.6)

For finite sequences (8.7) define (8 .8)

at/J =

[ 1]

-

[O J

E H0 !;0•

For finite sequences

J = (jo • i t

(8.9)

•

· · ·

)

,

i k � 0,

defme (8 . 1 0)

(3< 0 > =

[ I ] - [O J .

THEOREM 8 . 1 1 (LOCAL VERSION). As an algebra

where the tensor product is over all I, J as above and the coproduct follows by Hopf ring ' properties from the a s and {3's. o Anyone who knows H *K* can conclude 8 .5 and 8 .1 1 rapidly enough. We give this example to demonstrate the computation techniques in a familiar setting. PROOF oF 8.5. Just as in (7.38)-(7.40), a i) o a i) = 0. The relation e o e 1 = {31 is ( ( 1 obvious. All even generators must be truncated of height one because of Hopf rings and the

BROWN-PETERSON HOMOLOGY

53

fact that they are in H .K 1 and H* CP '"' . There are no additional relations in 8 . 1 1 , so 8.5 follows. o Homology suspend {j i) to define

(

(8.12)

�� E H2 (p i-1 ) H,

and a i) to define ( (8.13)

From 8 . 1 1 we have (8 1 4) .

THEOREM 8.15 (MILNOR [Mi2 ] ) . The coproduct on the Hop[ algebra H* H is given

by

n 1/l (�n ) = L

�0

·

�� ® ��

and

1/I (Tn )

=

n

L �� ® T; + �0 ·

Tn

®

1.

o

PROOF. We have our usual (stable) maps

l

MU = MU -- H and commuting diagram. From this we see that b = !;r � o b ; E MU.MU reduces to � = "1;1;; �� E H.n. (Recall that since {31 , i * pi, is decomposable it suspends trivially to H.n. )

0

Theorem 1.48(e ) now gives us the coaction o n CP '"' , just by the reduction MU.MU -+ H.H. Part (d) reduces

1/l
.L �pi ® �r

j;; o

For the T's we compare with the rs using the maps

This gives us all of the coproduct except the term ? ® 1 , which must be T ® 1 because we are in a Hopf algebra. To be fair we must reduce the entire proof of 1 .48(e) from MU to H; easily done . o The Tk and �k are seen to be Milnor's.

W. STEPHEN WILSON

54

PROOF OF 8.1 1 (RAVENEL-WILSON). This proof is by induction on spaces but we wil do al steps at once. We use the bar spectral sequence as in § 7. First we compute

E;, * = Torlf,".!S * (Z/(p) , Z/(p))

�

H* K* + t ·

Tor preserves tensor products and depends only on the algebra structure. Tor E {x) � r(ox) , r(y ) the divided power algebra, free over Z/(p) on 'Yn ( Y) with coproduct

llt('Yn ) = L 'Yn -i ® 'Y;; product 'Y;'Yj = (i, j)'Yi + i generated by 'Y(n = 'Yp n · We have 'Y t (y) = y and a is homology ) suspension. We have ox E Tor f"5�l (Z/(p) , Z/(p)) 'Yn (ox) E Tor!:�1 1(Z/(p) , Z/(p)). Tor TP t {x ) (Z/(p) , Z/(p)) � E(ox) ® r(l{)(x)) , with ox E To{�:J1{x)(Z/(p) , Z/ (p)) and the transpotence ,

and so

represented by the

1/)(x) E Tor;�,!7\Z/ (p) , Z/(p)) cycles x i lx p -i in the bar resolution , E� *. n , np lx i (Z/(p) , Z/(p)) . 'Yn (IP(x)) E To r2TP1 (x)

Thus

Since suspension is j ust o multiplication with

e 1 , this is

+

®E(e l o:I{jJ) ® r(l{)(o:I{jJ)) ® r(o:Il 4 o ) . Define

s(I) = (i 1 + 1 , i2 + 1 , . . . ) and s(J) = (O, j0, j l ' . . . ) . CLAIM 8 . 1 6. Modulo decomposables, in the spectral sequence pairing 7.24, for

(a)

and for

(b)

BK l X K *- 1 - BK ••

BROWN-PETERSON HOMOLOGY

55

We will flnish the proof of 8 . 1 1 before we prove the claim. For degree reasons the spectral sequence for BK0 collapses and we have already named the elements 'YP ;(t/>( [ 1 ] - [OJ )) = cr(;) ·

Letting S 1 -- K1 and comparing the spectral sequences for CP"" = BS 1 that the one for BS 1 collapses trivially, and so, like the cr(i) , the 'YP ;(f3(o ) )

--

BK1 , we see

= f3(i)

are permanent cycles. Thus by induction, the elements on the right in 8. 1 6 (using the dif· ferentials in 7 .24) are also permanent cycles. Since these are all of the even degree genera tors, the spectral sequence collapses. (The odd degree generators are in Tor 1 and so are permanent cycles.) Furthermore, since we have names for1 the elements on the left in 8. 1 6, ; we see that 'Y ;(cfJ(cr1{31)) represe � ts cr( i) o cr,i 1 ( ) o {38 + (I) and 'YP ;(cr1{3 J+ A o) represents P + J ) {3 i) o cr, ; ) o {3 8i(l ) = cr,; o (3'' (l + A o . We have now located and named all of the ele ( (I) (I ments

This is just an obscure way of writing e 1 o o:1 o {31 , o:1 o {31 without cr
0 = p * : H* K* -- H* K* . To show there are no extension problems all that we need to do is show (o:1

o

* {31) P

=

0,

but this is just p *(o:,(1) o (3'<1>). However, since we are trying to demonstrate Hopf rings we will give a Hopf ring proof. Each a1• o {31' can be written as ± o:{i) o o:1 o {31 or o:1 o {31 o f3( ;) · The proof is the same for both. By the distributive law (7. 1 4) (as in (7.42)), * (o:1 o {3 1 o f3( ;) ) P = o:s (I) o {3 s (l) o ( {3(�) ) = 0 . But by the same calculation, if f3c�) were not zero (such as in K(n) * K * ) this Hopf ring tech· nique would have solved our extension problem for us. o PROOF OF CLAIM 8 . 1 6. There is nothing to prove for the i = 0 case of 8 . 1 6(a). For i > 0 compute the p i - 1 times iterated reduced coproduct. The symmetric term from the right-hand side agrees with that on the left as cr1l+ A o . In fact, there are no other terms but the symmetric one. The terms in Tor ; with this symmetric term equal to zero are p easily seen to be the decomposables. Thus the two sides of (a) are equal modulo decomposables. When we establish the i = 0 part of (b) , the i > 0 part will follow in the same way. It is this i = 0 part that makes the odd primes a much better example of the spec tral sequence pairing than p = 2. Part (a) was done just by using the homology suspension

W. STEPHEN WILSON

56

fact that e 1 o e 1 = {3 1 • Nothing more complicated ever happens for p = 2 because there is never the fltration 2 transpotence to worzy about. We now get to use the r = 1 computa· tion of the pairing in 7 .24. Using 7 .24, iterated coproducts, the distributive law and the facts that, in our cases, [ 1 ] o x = x and [0] o x = 0, we have a(J) a(J t/>( (1 ] - (0] ) o C'i.a(I)p ) = (( [ l ] - [O] ) *P - I I [ 1 ] - (0] ) o C'i.a(I)p = (C'i.1 p l) *P - 1 1C'I.l p l plus many terms with more * products involved. We know what all cycles representing non· trivial elements in Tor2 look like. They are products of elements in Tor 1 or trans potence elements. Thus the terms with more * products contribute nothing. o 9. Two fonnal groups and BP • · In this section we bring in the second formal group law to add even more structure to the Hopf ring E. g •. We assume that both E and G are complex orientable (see after (7.25)) so we have xE, x 0 , pf , and P1° . We have al ready done a crucial step with one of the formal group laws by computing the multiplica tion in E. CP "" in 7.29. We need to define a few elements. For

(9.1 ) define (9.2) To make your confusion specific, this b1 stabilizes to For

b1_1 for E = MU = G.

(9.3) define (9.4) We now have a sub-Hopf ring (9.5) where E . [G * ] is the "ring-ring" (a group ring with some extra structure) on G * . We define a "formal group sum" (9.6)

Z

+ [ Fa )

y

=

*

i, j

(atlJ

o zol

o yo /

using the Hopf ring "addition" and "multiplication". We are ready to give the "main relation" which uses the interplay between the two formal group laws and Hopf rings to give unstable information .

1

THEOREM 9.7 (RAVENEL-WILSON (RW1 ]). Let b (s) = '1;1;; 0b1s • In E.Q. ( [s, t) ) ,

b (s +FE t) = b(s) + l Fa l b(t) .

o

BROWN-PETERSON HOMOLOGY

57

REMARK 9.8. This does not require the Kinneth isomorphism and so is not neces sarily in a Hopf ring. REMARK 9.9. For some E and G, such as K(n).K•• this is vacuous because b 1 = 0, i > 0. REMARK 9 . 1 0. The notation is almost necessary because when 9.7 is expanded out we have

(9 . 1 1 )

P·(?r·t."··)'

=

; 1a31

•

(P··· ) ( pm •m )" "

'

.

and the actual coefficients of s 1ti are even more unpleasant. G PROOF OF 9 .7. We compute the composition ep• x ep• .. ep• � two different ways. b(s +p

g2 in

t) = xf([j(s +p t)) = xf((j(s){j(t)) = xfp.((j(s) ® {j(t}) = (x 0 o JJ) . ((j(s ) ® {j(t)) = (p * (x 0)).({j(s) = (J; a1y (x ft ® (xf)i ) ({j(s) •

1/

=

*

ij

[a1y J o

® (j(t))

® {j(t))

·

b(s)ol o b(t)o i = b(s) + ( FG ) b{t) .

The second to the last step is because the sum goes to to o . o We obtain the following corollary by iteration.

*•

a1i to [a1i] and the product

the

COROLLARY 9 . 1 2 ( RW. J . ln E * g * [ (s ) ) , o

b ([n ] p8 (s)) = [n ] l F J (b (s)). c

Again, as with 7 .29, this appears to be a useless nonsense formula; but again, as in 7.32, when we restrict to BP we can obtain very useful explicit results. To. do this we go to the mod (p) homology . THEOREM 9 . 1 3 [RW1 ) .

Let 1 = (p, v1 , v2 ,

• • •

).

In

2 QH. BP2/ [I) 0 QH * BP*,

PROOF. Using 9 . 1 2 for n = p we have

b o = b {ps) = [p ) ( F ) sp (b(s))

=

L (FJ [vn ] o b(s'f P

n >o

n

* [p) o (z),

by 3. 1 7.

Since [p] o (z) = b0 mod decomposables, this becomes, mod decomposable& and [I) 0 n n l;n > O [vn ] o b (stP . Picking out the coefficient of s P gives the result. o

2

,

W.

58

STEPHEN WILSON

We still have not answered the question about how powerful the "main relation" 9.7 is. In the universal case G = MU, it is complete and gives all information. 2 * (E*BP2 *) is generated over E * THEOREM 9 . 1 4 (RAVENEL-WILSON [RW 1 ] ). E * * by [MU J ( [BP ] ) and the b's (b m 's). The only relations come from 9.7. o The proof is difficult and will not be given here. To obtain

(9. 1 5) it is enough to add e 1 E E 1 MU1 (E 1 BP1 ) and the relation

(9. 1 6)

A key step in the proof is to show that the integral homology of BP* has no torsion . With this the Atiyah-Hirzebruch spectral sequence for BP * (MU*MU*) collapses. The above solves all * and o product extension problems. By duality we have a complete de * * scription of MU MU* (BP BP* ), the unstable operations. The last section is dedicated to developing this further. The explicit formula 9 . 1 3 can be used to give a basis for QH*BP* and PH.BP* which then lifts to QE.BP* and PE* BP* . E *BPk is an exterior algebra or a polynomial algebra for k odd and even respectively. Let (9. 1 7) and

(9. 1 8) both nonnegative finite sequences. Define

(9. 1 9)

[v IJ b J - [ v1i t v2i 2 _

• • •

J o b (oioo)

o

b (oit t) o

• • •

•

THEOREM 9 . 20 [RW1 ] . A basis for QE.BP2 * is given by all [v 1] b J such that if J can be written as

J = pAk t + p 2 Ak + · · · + p n Ak + J ' ' 2 n

J ' a nonnegative sequence, then in

= 0.

o

This basis uses 9. 1 3 to get the largest v's possible . This is useful in the study of the BP k [Si] and the unstable Hurewicz homomorphism for BPk [Ma] . It is also p ossible to use 9 . 1 3 to obtain a basis using the smallest v's possible. This basis is particularly useful for computations with unstable o perations; see § l l .

THEOREM 9 .2 1 (BOARD¥AN). A basis for QE*BP2 * is given by all [v 1J b J such that if I can be written as

I = Ak o + Ak t + . . . + A kn + I ' , k I' a nonnegative sequence, then in < p n . o

BROWN-PETERSON HOMOLOGY

59

For BP .BP* ' the b i) are not always the best elements to work with. Stably, in ( BP* BP, the t; are not everyone 's choice either. The elements c(t; ) will certainly do as alter native stable generators. Even better, they desuspend to elements

f; E BP2 P ;BP2 .

(9.22)

For this reason many people prefer them. Thus in 9.20 and 9.2 1 the b i) can be replaced ( by the f; (by 3. 1 4). 1 0. Chan's proof of no torsion in H.BP •. As already mentioned, knowing there is no torsion in the homology of BP* is very important. The first proof of this fact in [W 1 ] is very ugly and we suspect no one ever reads it. There the spaces BPk are dismantled in an unpleasant way. Then every possible torsion element that could arise during reassembly is hunted down and killed with such individuality that the process borders on sadism. The re sult, even in this weak form, already has several applications. It allows one to study the en tire homotopy type BPk . The important spaces here are the BP
of

· · ·

'

,

THEOREM 1 0. 1 ( [W1 ) ; SECOND PROOF [RW1 ) ; C HAN S PROOF [Ch] ). H* BP2 k + l '

is an exterior algebra and H* BP; k is bipoly nomial. We have, for k + n > 0 , rank Qlk + n BP� = rank 7Tk BP = rank HkBP.

o

S KETCH OF C HAN S PROOF. The proof is by induction. To ground the induction we '

have

which proves the result for degree 1 = k + � i < k, all n, and Hn + kBP� . the bar spectral sequence

Hn +iBP� , 0

n. For our induction we assume the result for n < q . We must show it for Hq + k BP� . We use

W. STEPHEN WILSON

60

E�

*

=

Tor:: BPq -1 (Z/(p) , Z/(p)) .,. H* BP; .

For q odd, Tor of a polynomial algebra is just an exterior algebra on the suspensions of the generators. The spectral sequence obviously collapses and we are done . F or q even, Tor of the exterior algebra is a divided power · algebra r on the suspension of the gener ators. For both odd and even cases we compute k+qk+q - IEt, t

=

Tor,, t

t known below this line

=

k + q

For q even, r is even degree so the spectral sequence collapses (in our range , by the diagram). Since r is dual to a polynomial algebra we have the cohomology part of bipolynomial . The rank of QJI *BPq is also correct. We must now solve all extension problems i n E 2 = E oo = r. We must show that it becomes a polynomial algebra of the correct rank. Since its dual is a polynomial algebra of the correct rank we know that i f it is a polynomial algebra then its rank will be correct. Let A be the algebra after the extension problems are solved. If A is not p olynomial then clearly rank QAk + q > rank 1rkBP. By induction everything is okay in lower degrees and we are only concerned with degree k + q. . Now, use the bar spectral sequence to compute Hk+q + i BP; + i , i > 0.' (By induction we already know the lower degrees.) It is easy to see that this oversized rank wil persist into the stable · range, which is a contradictio n because . we know the stable homology. Thus ·

A must be a polynomial algebra of the correct rank.

o

Th is technique o f "trapping" the homology o f an Sl-spectrum between the known in the form of H0 g_ * and the known H.E. has wide applications. This is the easiest case. It can be used in other instances to solve extension problems, to show collapsing, or to compute differentials. 1r.E,

Part III. Something New 1 1 . Unstable operations. When I first showed, in 1 97 1

[W 1 ] , that H* BPk had no

torsion, I realized it meant that unstable BP operations were accessible and abundant. It was hopeless to study them until a basis for H* BPk had been found [RW 1 ] . It is only in recent years that I have d abbled seriously with them. The usefulness of the only other truly unstable , additive operations, the Adams 1/l k in K-theory , led me to believe the BP operations carried a great deal of information . The recent work of Bendersky, Curtis, Miller and Ravenel (see [BCM] and later works) supports this conclusion . Since K-theory splits off cobordism [CF 2 ] , we know that the unstable cobordism operations contain, at minimum, the same in formation as K-theory . However, a quick look at the homotopy type of BPk [W2 ] makes it clear that K-theory operations are only the surface of the operations available to cobor dism. Although it is clear that BP operations contain a great deal of in formation , the old problem with BP is still there : how do you get information out and live to teii the tale? The machinery is now set up for straightforward computations, but, as yet, I have been un able to do a general computation with applications of interest. This has caused a delay in publishing these results. The reader has mercifuiiy been spared the tortuous backwaters that months of calculations led me to. They have resulted in only a few conjectures to which I hope to return someday. First I will turn the unstable operations into a "ring" and give a rigorous general non sense definition of unstable modules, which is somewhat independent of the rest of the sec tion . Second, I show that unstable operations are not just a simple divisibility problem for stable operations as I first suspected, but that unstable operations are very different animals despite their close relationship with stable operations. More explicitly, no multiple of a truly unstable operation is ever stable ; although rationally , unstable and stable are the same ! This is our main result. It is a fact first revealed by unspeakable computations of examples. This property is demonstrated in a familiar setting by showing that the Adams operations for complex cobordism are legitimate unstable operations. Just as with K-theory, 1/l k can be delooped twice for every power of k it is multiplied by. In order to prove this , formulas for computing {1 1 .1} are worked out for all unstable operations r E BP kBPn = [ BPn , BPk ] . The proofs i n this sec tion allow us to talk about stable operations, an aspect of BP neglected so far in these lectures. 61

62

W. STEPHEN WILSON

Near the end of this section I have presented the gruesome details of an elementary application. The motivation is to show that straightforward calculation does produce, but this is not the way to do it. It is included for the student who really wants to learn the material. The research related to this section took place over the years at The Institute for Ad vance d Study , Oxford University, The Johns Hopkins University, and the Centro de Investi gacion del I.P.N. in Mexico City. In addition to the support of these institutions, the research was partially supported by the National Science Foundation and the Sloan Founda tion. I am grateful to Luis Astey, Mike Boardman, Daciberg Goncalves, Don Davis, Vince Giambalvo, Sam Gitler, Dave Johnson, Peter Landweber, Dana Latch, Haynes Miller, John Moore, Bob Thomason, and surely, a few others I have forgotten ; al of whom helped me out in one way or another. Since this work was completed, Mike Boardman has developed another approach to unstable operations and he can reproduce most of these results and more. In particular he has proven the conjecture of mine that QBP.,.BP.,. is [BP.,.] free as well as BP.,. free, a curious fact indeed . We have described BP.,.BP.,. which gives, b y duality, one description of the unstable · BP operations. For our definition of unstable modules, we will go to a more general setting. Let

{1 1 .2)

* E (-) � [-, g.,.)

be a generalized cohomology theory. The unstable cohomology operations are the natural transformation k E x - E nx

I

( 1 1 .3)

which, by [Br] , are given by

{ 1 1 .4) We wil restrict our attention to additive operations:

{1 1 .5)

r(x + y) = r(x) + r(y ).

To do this effectively we need to assume that

(1 1 .6) The "abelian group" structure of Ek •

(1 1 .7)

BROWN-PETERSON HOMOLOGY which comes from the abelian group

E kX,

63

gives rise to a "coproduct":

( 1 1 .8) and the "primitives", PE*!Ik • i.e., those r E E *!Ik such that r -- r ® 1

( 1 1 .9)

+ 1 ® r,

are the additive unstable operations. We want an unstable E*E module to behave like E *X for X a space . That is, we want a graded (topologized) E * module, like E *X, and pairings like

(I l . I O) giving a commuting diagram

(1 1 1 1)

where the composition of maps gives the pairings ( I 1 . 1 0) and { 1 1 . 1 2) We take the time now to make rigorous what we mean by the "ring" of unstable (ad ditive) operations and modules over it. This part can be safely skipped by readers who need to do o . It is understood that upper * modules are topologized and that maps occur in the ap propriate category. This will be suppressed throughout. We have R algebras

s

E * = E_ * ,

( 1 1 . 1 3) A graded E *

- G*

G * = G- * '

etc.

bimodu/e is a bigraded R module

( I 1 . 1 4) where s; is a left E * module, B� is a right G module, and these structures are compatible * in the obvious way . We denote this category grbimod£ *- G • . We need a "tensor product" functor ( 1 1 . 1 5)

( 1 1 . 1 6)

64

W.

STEPHEN

WILSON

the zero degree of the standard graded tensor product over F* = F_ •• i .e . ,

(1 1 . 1 7) One is not allowed to call just anything a tensor product. Bob Thomason has justified call ing this a tensor product by showing that it is a coend , as any good tensor product should be . The of a functor [ML, p . 222]

coend

(1 1 . 1 8) is an object,

(I 1 . 1 9)

S: cop

X

C -+ D,

d E D, and a dinatural transformation S(c, c) --+ d 71:

such that if X E D has a dinatural transformation

S(c, c) --+ X, then we have a: d X with = To be dinatural means that for a: c -+ c', the following diagram commutes: S(c, c) (1 1 .2 1 ) S(c', c) � � �S(c', c') ( 1 1 .20)

fj:

a71

-+

/3.

Let C = F*, where, as a category , F* has objects

Fi-i = morph(i, j).

(1 1 .22) C0P

i E Z and morphisms

= F with Fi-i = Fi -i *

=

morph(i, j); Define

(1 1 .23) The coend of this functor is just

(1 1 .24) as defined above. Let

(1 1 .25)

B; E grbimodE * -F • ' c; E grbimodF * -G •

Define the graded E *

(1 1 .26) by

(1 1 .27)

-

F bimodule *

and

D; E grbimodE * -G • .

BROWN PETERSON HOMOLOGY

65

The left E * module structure is inherited from the left E * module structure of v:. The right F module structure is induced by the left F * module structure on c:. * The adjoint relation is {1 1 .28) Let o: E grbimodE * -E . A map •

( 1 1 .29) puts a "ring" structure on n:. The definition of associative is obvious since ®E . is associa tive. A unit is a sequence of elements I n E n; which act as left and right units in the ob vious fashion. When we say ring we mean an associative "ring" structure with unit. Observe that this is a sequence of maps (1 1 .30) EXAMPLE 1 1 .3 1 . End!(M*). Let M* be a left E * module. Let

The left E * mod ule structure of M* turns End!(M*) into an object of grbimodE * -E . The • "ring" structure is just c ompositio n of maps and so it is clearly associative. The unit is the collection of identity maps I n E homR (M n , M n ) . More generally, we can define, for M: E grbimodE * - G • ,

which is again a ring in grbimodE * -E . The first is a special case of this. o • EXAMPLE 1 1 .32. 0: MODULES . A 0: module, M: E grbimodE * -G • , for 0: a ring in grb imodE * -E , is a ring map •

which, by the adjoint relation, is the same as a definition given by an E * with commuting diagram

As above, we can specialize to the case of a left E * module M * .

o

-

G * map,

W.

66

STEPHEN WILSON

·

* EXAMPLE 1 1 .33. STABLE E E MODULES. let E *E be the stable cohomology oper ation ring for a generalized cohomology theory represented by the spectrum E. Define St : E grbimod * - by

E E•

en- kE = St� .

This is not a very efficient way to say it, but a ring map st: -+ End:(M*) is the same as our usual concept of a stable E *E module structure on M*. There is a stable E *E module structure induced on each I; 1M*, i E Z, from st : � S t: ::

End :::(M* ) � End:(I;1M* ).

-+

o

EXAMPLE 1 1 .34. UNSTABLE E *E MODULES. This is our motivating example. Let E = E.. be an n-spectrum with E *( E -k

X

E -k -k � E *E -k ) � E *E

for every k. We have PE *E.k C E *E,k . Let u: = PEnE,k . This is a ring in grbimodE * -E . • We define an unstable E *E module structure on M* to be a ring map

u:

-+

End:{M*).

The cohomology suspension en- kE -+ PEn E." gives a ring map st: -+ u:. By composi tion, any unstable module has a stable structure on it. We say a stable E *E module is un. stable .if the following diagram can be filled in : St :

!

u:

-+

//

End:{M: )

///�

A

stable module may have many unstable structures, or none. An unstable module structure on M * can be lifted to an unstable structure on I;1M*, i ;> 0, by using the cohomology suspension map PE *E.. -+ PE *-1E,._1 to get

u:

-·

u::: -+ End :::{M * ) � End :(I; 1M) .

Extending the unstable structure on M * to one on I; -1 M * (or I; -1M*) may not always be possible. This, of course, is one of the important points about unstable operations. Every E*X, for X a space, is an unstable E *E m� dule, and E*I;1X is an unstable module isomorphic to I;i'f*X. If a space can be desuspended , say X � I;Y (Y = I; - 1 X), then {1 1 .3 5) and we have an unstable strtJcture on I; - 1£*X. Therefore, if there is no compatible un stable structure on I; - 1E*X, then X cannot be desuspended . We have already noted that for E = MU and BP, {1 1 .36)

BROWN-PETERSON HOMOLOGY

67

and (1 1 .6) holds. This is simply because H* §.* has no torsion and the Atiyah-Hirzebruch spectral sequence collapses. Standard mod (p) homology also satisfies condition (1 1 .6). The stable operations H *H give the Steenrod algebra A P . We can exhibit the striking differences between unstable BP and H operations and show the reasoning behind our philosophy that BP operations should contain a great deal of information . We have, for n > 0, A p - PH *-Kn C H*K - n primitively generated, {1 1 . 37) BP* -nBP >-+- PBP *BPn C BP*BPn not primitively generated. For mod (p) cohomology, the stable operations map onto the additive unstable opera tions and H *Kn is primitively generated. Thus the only unstable operations are stable oper ations and cup products. The main unstable information is that some stable operations (for p = 2, Sqi, i > n ) are always trivial on n dimensional classes. For BP, the stable operations inject into the additive unstable operations. As a bonus, BP *BPn is not primitively generated . So not only do we have a rich new collection of ad ditive unstable operations, but there are serious new nonadditive operations not generated by cup products and the additive operations. We will tend to ignore the nonadditive operations until useful applications of the ad ditive ones have been found. Theoretically, though, it is clear how to deal with them, and computations like those in this section can be carried out. Recalling that H MUn is bipolynomial or exterior, we have MU MUn is cofree and * * MU*MU is either a completed polynomial algebra (n even) or a completed exterior algebra * (n odd). We can see by duality that MU*MUn is not primitively generated . Since there i s never any torsion anywhere i n BP (MU) we can feel free t o study things rationally without loss of information. BP is just a bunch of Eilenberg-MacLane spaces ra tionally so we have ( 1 1 .38)

BPQ*-nBP � PBPQ*BP -n•

n > O,

with obvious modifications for n ..; 0. The isomorphism is given by cohomology suspension. This immediately implies ( 1 1 .39)

0 -+ BP * -n BP -+ PBP *BP -n•

n > O,

with modifications for n ..; 0. From this it appears that the study of unstable operations for BP is just a problrm of divisibility conditions for stable operations, and indeed, if we restrict our attention to k dimensional complexes this is the case, and the coker defined by (1 1 .40)

BP* -nBP -+ PBP *BP - n -+ coker -+ 0

would b e entirely torsion. However, a s w e let k go t o oo, w e have a n inverse limit with higher and higher torsion. (Think p-adics.) In other words we have completion problems, discussed more later, and unstable operations cannot be represented in terms of stable opera tions and divisibility problems, but they are truly different objects. Our main theorem is THEOREM 1 1 .41 . In (1 1 .40), there is no torsion in the cokernel.

o

W. STEPHEN WILSON

68

We defer the proo f for a bit. We have : n BPQ_- BP

(1 1 .42)

=!

:: BP

* n - BP � ho m8p • (BP.BP , BP *)

n

n

We now have two ways to describe o ur unstable operations. First , they are d ual to n, which we have an explicit basis for . Second, we know they are contained in the QBP ra tional stable operations . For this to be a usef ul statement, we must know how they s it inside. Our first description can tell us. We have

(1 1 .43) * and the unstable operations, PBP BP n , in

(1 1 .44) a re just those ma ps which send QBP n to BP * tion of the isomorphism (1 1 .43). n We can thin k of rE BP k- BP as a map

C BP

�. We need a more explicit descrip

(1 1 .45) j ust by restricting to n dimensional classes (the cohomology suspension again). The map in (1 1 .43) is induced as usual: r goes to

( 1 1 .46) with

(1 1 .47)

r(y) = (r, y} or r(y) = € 0

S00

0 r.(y),

homology suspension an d r* the i nd uced map ( 1 1 .1). Although o ur concern is mainly r, evaluating r* i s a n interesting problem i n its own

S00 the oo

right. As mentioned already , i t is enough to do this rationally. Recall from §9 the notations o , * • [a ] , b = �r>o b; and xE BP2CPoo. The algebra * str uct ure on BP.BP induces a coalgebra on BP BP with r � �r ' ® r". n THEOREM 1 1 .48. Let rEBP �- BP C PBP �BP n , any n (also for MU). Jf r(x) = k �;;;. o ; X(( -n )fl) + i + l , C; C EBP S. then * (i) r* (y z ) = r. ( y) * r (z), r(x * y) = 0, (x, y) in the augmentation ideal;

* r * (y 0 z) = � r '(y ) 0 r "(z), r( y 0 z ) = � r '( y ) r" (z); (iii) r . ( [a] ) = [r(a ) ] , r( [a ] ) = r(a); (iv ) r *(b ) = *; ;;. o [C; ] o b 0 (( k -n) /Z ) + i + l , r(b) = � k;;> O Ck = C. 0 This gives a complete evaluation of '* for rE PBP �BP n because we know that the [a ]

(ii )

* and b's generate BP.BP n from §9. Since BP BP n is primitively generated ration ally, it is easy to extend this to all rE BP kBP n.

69

BROWN-PETERSON HOMOLOGY

PROOF. (i) follows from the additivity of r and the fact that homology suspension is trivial on decomposables. (ii) and {iii) are elementary. To show (iv), we consider, rationally,

Then

From

e(b) = 1 , s(x y) = 0, and e(17R (a)) = a , we have *

r(b) = e 0 s "" 0 r.(b) = e 0

s""(

* [c; ]

i>O

0

b ((k-n)/l )+i+ l

)

We are now equipped to produce some honest unstable operations. If we are given

( 1 1 .49) we

can evaluate it on QMU.MUn , and if it lands in MU* C MU� we have a legitimate un stable operation. Several people have studied rational Adams operations for MU and BP, but generally they are not viewed as potential unstable operations [N, Ar2 , Sn] . Recall {1 .53) that the MUQ multiplicative operations are given by power series

/{x) = X + . . .

{1 1 .50)

with coefficients in MU� . Let ( 1 1.5 1 )

f(x) = [k] (x )/k

define a rational multiplicative operation 1/1 k. We know that for x E MU'& CP,. ,

(1 1 .52)

1/J k (x) = [k] (x)/k.

THEOREM 1 1 .53 . .p k e PMU0MU. -0 MUg MU. E = 0, 1 , i E Z. o

Q

- 1 •+ e C,

REMARKS 1 1 .54. Novikov [N] claims the first part for torsion free spaces in a foot note. The result is true for BP as well because the y, k exist there [Ar2 ] . Note that the .p k 2 imply torsion in the coker of PBP *BP• ,._. PBP * "" BP. _1 • 1 PRooF. Recal (3.7) that mog x = log / - (x). We have / - 1 (x) = [ 1 /k] (kx) because

[ 1 /k] (k([k] (x)/ k) = [ 1 /k] { (k]{x)) = x.

70

W.

STEPHEN WILSON

So mog x = log f - 1 (x ) = log [ l/k] (kx) = log ex p(( l/k) log(kx)) =

and since

(1/k) log(kx) = (1/k) L m;(kx)i + t , L 1/l!(m;)xi + 1 = (1/k) :L m;(kx)i+ 1 , ;;; o

we have 1/1 !(m;) = k ;'fn:; and it follows that, for a E Mu-u,

1/l k (a) = kia.

0 We will prove only the MU MU0 part as it adequately demonstrates the techniques. Recall Theorem 9. 1 3 that MU.MU is generated by b 's and [MU*] . So QMU.MU0 must be gen *

erated by elements of the form

[a] with lal

=

o

b :i t

o

b ;h

·

· ·

= [a ] b J

- 2{i 1 + i2 + · · · ), [a} E MU0MUi · We have al

[k] ( 1 )/k . Since 1/1 k is multiplicative we have

$ k [a] = k - la l 1 2 a and $ k (b) =

$ k ( [a J b J) = $ k ( [a] ) � k(bJ) = k -la l f2a $ (bJ) =

k - la lf 2 a(stuff)/ki 1 +h + ··· ,

which is integral! Likewise for the general case . o Before we prove 1 1 .4 1 we introduce the stable BP operations. Define monomials in the t's in BP.BP by ( 1 1 .55) We define ( 1 1 .56)

rg E BP *BP

by the property (1 1 .57) Then lrg l = lEI = 2 'J;;;; o e;(P ; - 1 ) . Let R be the free Z(P ) module on the 'E · R is a co algebra with (1 1 .58)

rg -+ L 'E' ® rg " E ' + E " =E

and ( 1 1 .59)

BP *BP

=:

BP * ® R.

An element r E BP *BP is a possibly infinite sum ( 1 1 .60)

r = L eEtE ,

cE E BP * .

BROWN PETERSON HOMOLOGY

71

Our interest is in BPQBP, which is not the same as tensoring BP *BP with Q, but is the same as taking sums ( 1 1 .60) with the cE E BPQ . Define a truly unstable operation to be an element of PBPkBP which is not in the image of Bp k -"BP, i.e., an additive unstable opera " tion which is not a stable operation. We can rephrase 1 1 .4 1 . THEOREM 1 1 .6 1 . A truly unstable operation r = !:,cE rE must be an infinite sum unbounded negative powers of p in the coefficients cE . o

with

Define af E BP2 i by ( 1 1 .62)

LEMMA 1 1 .63.

rE (x) = L afx 1E I / 2+ i + l , Let a E =

x E BP 2 CP• .

;;;.o

'I:,;;;. oaf.

Then

PROOF.

Since (x, (j1> = 0 except when j = 1, we have

the required result. The first equality is just 3 . 1 4. o Tables of some af for p = 2 have been computed and are presented at the end of this section . Thanks to Daciberg Goncalves and Don Davis for tremendous help. C oROLLARY 1 1 .64.

rE (b ) = a E .

CoROLLARY 1 1 .6 5 .

.,. A;

0

A·

r (b ( l) ) = a0 ' = - 1 .

PROOF. Use induction and

o

"T, Ft1 c(t1 ) P1 = 1 . o

E PROOF OF 1 1 .6 1 . Let f = !:,d rE E PBP *BPn C BPQ -"BP not in BP*-"BP C PBP *BPn ; then 0 =I= d E E BPQ /BP * for some E. We assume that p kf = r E BP *BP for some minimal k > 0. We obtain a contradiction. This proves the result. Since we have multiplied f by p to get r, r must be trivial mod (p ) . By cohomology suspension, f is in .al

W.

72

STEPHEN WILSON

PBP *BP;. i <; n, and the corresponding f's are also trivial modulo (p). Since we hav� chosen k minimal we can write r

=

E cErE + pr ',

some r' E BP *BP

with cE 1- 0 (p) and some nontrivial c E . The pr' will automatically have pr' = 0 mod (p) . So since ; is zero mod (p), we must have r" = !:,c ErE give r" = 0 mod (p). We show this does not happen, i.e., for any such r " with cE ¥ 0 mod (p) we find that for large negative i, r" is nontrivial mod (p) on QBP .BP;. Of course we need to find only one element X with r" (x) 1- 0 (p). Order the sequences E by

£1 < £2 if 1£ 1 1 < 1£2 1 , £ 1 < E2 if 1 £ 1 1 = 1£2 1 and /(£ 1 ) > 1(£2 ) , l(E) = ,L e;, £1 < £2 if 1 £ 1 1 = 1£2 1 , 1(£ 1 ) = 1(£2 ) ,

e/ = el, i = 1 , 2,

. . . , k - I , and

e�

<

er

Pick the minimal E from the set of cE. To construct our x we begin with bE = oe2 . o f E, 1 1 .65 and 11 .48, b ( t ) o b ( 2 ) o . . . By our ch mce oe l

(I 1 .66)

We started with a fixed PBP *BPn and we must produce an x in P,BP *BP;, i <; n. This bE may not meet this condition. Pick k and j very large so that [vf'] o bE E PBP *BP;. i <; n. Also choose k bigger than any v in cE, and j bigger than 1 (£) and I eEl ; then (1 1 .67)

�

.

cEr ( [vfJ] E

o

.

bE) = ± cE vfJ

(p).

For � c ' 'E , > E, to give something nonzero we know that some of ' ' must apply to [vf ] , E E by (1 1 .66). I f so, then it must apply to all pi of the vk 's in order to be nonzero mod (p ) . This guarantees pi v's smaller than vk . However, by our choice of j, cE in (I 1 .67) has fewer such v's, so the term in (1 1 .67) will never be c�nceled out. o This gives some insight into the number and type of unstable operations. We can take a "basis" for PBP *BP* dual to either 9.20 or 9.2 1 . For the element r corresponding to a basis element (I 1 .68) the "lead" term will be ± rE/pl(I) for some E. The infinity of terms in the "tail" of r will be forced to be there in order to make r integral. In practice it is a burden to handle the entire definition of unstable operations; a "local" condition does well enough. If we have a stable BP *BP module M * , then for any n x E M we have a map BP * -nBP - M* (1 1 .69)

i

BROWN-PETERSON HOMOLOGY

73

which takes r to r(x). If M* has. an unstable structure on it, then this map must .factor through a BP*BP module map

( 1 1 .70)

This is just good old-fashioned algebraic topology. The BP*BP module structure of PBP*BP11 is complicated. Basically there will be a bunch of generators £11 , y 1, y 2 , , and a set of operations and relations • • •

( 1 1 .7 1 )

Defining the map in (l l . 70) amounts to choosing values in M* for the unstable operations ( 1 1 .72)

which are consistent with the relations ( 1 1 .7 1 ). This is difficult to handle in practice, but it is always a straightforward calculation in primary operations. We give a computational example. The result is well known and can be proven using standard unstable secondary operations. The hope, of course, is that the unstable primary BP operations can eventually be used to obtain results that would necessitate the use of standard higher order operations. To make our computations, we formalize the previous dis cussion in the equivalent form: Is M11 in the image of (1 1 .73)

M* = homBP * BP(BP*BP, M * ) +- homBP* B P(PBP *BP11 , M * ) ==:

Ext:P * BP(PBP*BP11 , M* )?

We need to produce only one element x11 not in the image in order to show no unstable module structure exists. To show an unstable structure exists one must show that all of the local definitions patch together consistently. For our example, consider the cofibration ( 1 1 . 7 4) No eleven-fold desuspension exists for RP�:- We prove this by showing there is no unstable module structure on ( 1 1 .75)

which is compatible with the given stable structure . We compute, for p = 2, ( 1 1 .76)

W.

74

STEPHEN WILSON

and show that it is not onto M5 • It is enough to compute through internal degree 1 5 be cause M k = 0 , k > 1 5 . Also M k = 0 , k even. We describe a co free BP BP resolution * 0 -+ QBP.BP5 -+ F0 -+ F1 , { 1 1.77) take the dual and apply hom to obtain {1 1 .78) 0 -+ homBP * BP(PBP*BP5 , M* ) -+ homBP* B P(Fri. M* ) � homBP * B P(F: , M * ) .

To briefly describe how this goes (through deg 1 5), F0 :: 'I; 5 BP.BP e I; 1 1 BP. BP e I; 1 5 BP.BP { 1 1 .79) and

( 1 1 . 80) Now, { 1 1 .78) becomes {1 1 .8 1 )

where r is a matrix o f stable operations. Interpreted a s i n { 1 1 .7 1 ), for every x 5 we must be able to choose an x 1 1 and an x 1 5 such that { 1 1 .82)

0=

r 1 1 x 5 + r 1 2 x 1 1 + r 1 3x 1 5 and

0=

r2 1 x 5 + r 2 2x 1 1 + r 2 3 x 1 5 •

This wil not come out so clearly in the computation. What we want is to show that the composition of {1 1 .8 1 ), {1 1.83) 0 -+ Ker -+ M 5 e M 1 1 e M 1 5

i

Ms is not suJjective. We wil never describe the matrix r in terms of the 'E • but we will keep our computations in homology, not cohomology. From 9 .2 1 , or simple manipulations using 9 . 1 3 and 9 . 1 4, a basis for QBP.BP5 is deg 5 7 {1 1 .84)

9

15

e l o b 102

e1 o b 1 o b2 e l o b 20 2

e 1 o [v 1 ] o b 1 o b 2 o b 4 , e 1 o [v2 ) o b ; 3 o b ; 2 .

The Boardman basis is preferred because it uses the smallest v's possible .

75

BROWN-PETERSON HOMOLOGY By homology suspension we consider this a subcomodule of � 5 BP BP. Recall that * [a] suspends to 11R (a) and b suspends to b - .

n

n t

QBP.BP C � 5BP.BP 5 deg 5

(1 1 .85)

b�

9

From tables at the end of this section giving 17R (v 1 ) = v 1 - 2b 1 and 17R (v2 ) = v2 5vib 1 - 1 5 v 1 bi - 2b 3 + 1 4b � we can rewrite and slightly alter the basis to be

QBP* BP C � 5BP.BP 5 deg 5

(1 1 .86)

7

b1

9

b 21

11

b3 ,

2b 31

13

b lb3,

v 1 b� - 2b1

15

2b�b 3 , v 1 b1 - 1 4b�

+

76

W. STEPHEN WILSON

To begin our resolution deg

QBP

(1 1 . 77), (1 1 . 79), (1 1 . 80), in F0 we give coe fficients 5 I: 1 1 BP*BP I: 1 5 BP*BP 5 --. I: BP*BP

5 7 9 11 (I 1 . 87)

13

15 (v 1 b 41 - 14b 51 ) --. (v 1 b41 - 14b 15 ) This requires some computations to verify as does

Fo � F1 coefficients of deg

11

b 3 --

0

b 31 -1 u --. 13 (1 1.88)

15

-2

b 41 --

3

b 1 b 3 --

6

b 1 1 1 1 --

-2

b s1 --. b�b 3 --

- v21 + 1 7v 1 b 1 - 1 7b 21

bf l u --. 1 1 5 -M*

Before we can compute of (1 1 . 75) .

(I 1 .8 3 )

- I 7vf + 6 8 v 1 b 1 - 68bf - 33 v 1 + 66b 1 from

(1 1 .8 1) and ( 1 1 . 78) , we

must compute the groups

77

BROWN-PETERSON HOMOLOGY

We begin with the fibration

( 1 1 .89) The Atiyah-Hirzebruch spectral sequences all collapse and give an exact sequence, so

BP*Rr• � BP * [ [x] ] / [2] (x).

(1 1 .90)

Furthermore, the spectral sequences collapse for everything in sight, giving surjections from the maps

RP2 k - RPoo , RP oo 2k

(1 1 .9 1} where CP;

�

-

CJ'ao

k'

RP2m 2 k - CP"k •

P" /CPk - 1 • For example,

BP *RP 2 k � BP* [ [x] ] /( [2 ] (x), x k + 1 ) ,

( 1 1 .92) and

BP*RP22 " k

(1 1 . 93 )

has a Z< 2 > summand in degree 2k coming from x k E BP 2 kCP;" . In memory of CP .. we k call the BP* generators for (1 1 .93), x k, x k + 1 , . . . , x " . Using the formula for [2] (x) at the end of § 3 we can describe BP *RP;! explicitly from degree 1 6 on up.

BP *RNt = � 1 1M *

(I 1 .94)

deg

groups

generators

relations

26

Z/(2)

x1 3

2x 1 3 = 0

24

Z/(4)

x1 2

zx1 2 = v x 1 3 1

22

Z/(8)

x1 1

2x 1 1 = v x 1 2 1

20 Z/( 1 6), Z/(2)

x 1 0 , v2 x 1 3

2x 1 0 = - 3 v 1 x 1 1 + v2 x 1 3

1 8 Z/(32), Z/(4)

x9 , v x 1 2 2

2x 9 = - 3v 1 x t o - v2 x 1 2

16

xs • v x1 1 2

Returning to

(1 1 .95)

z< 2 >, Z/(8)

( 1 1 .8 1 ) we

have

0 - Ker - (Z< 2 > Ea Z/(8)) Ea Z/(8} Ea Z/(2) r=(r 1 •'2 ) Z/(8) Ea Z/(4) ,

with new names for the generators. Since we obtain surjections from (1 1 .9 1) we can com pute our stable operations in BP *CPJ 3 • This is torsion free, so we can compute operations in here by duality in BP. CPJ l . These in turn can be computed using CP00 • Our operation

W. STEPHEN WILSON 78 r is given to us in terms o f (I 1 .88), and we must compute it on the y's in (I 1 .95). r(x n ) = � a1x1,

For

(I I .96) which is convenient for us because we are given

(I 1 .97)

n

r in terms of r

•.

The map x n is just

n

We will need the following computations which use the fact that the tables at the end of this section.

b 2 = - v 1 b 1 + 2b� from modulo

x PtJ 1 3

=I

(2)

x � 1 tJ 1 1

=I

(8)

x � 1 tl1 2

(1 1 .98)

= I Ib 1 = -b 1 x P P1 3 = l lb 2 + (�1 ) b� = v 1 b 1 + b� x !tJ1 1 = 8b3 + (�) b � + ( I� I ) b1b 2 = 0

(2) (8)

x !tJ1 2

= 2b 1

(4)

x !P1 3

=0

(2) We compute in BP * CP� 3 but because we reduce as in the tables Let

(I I .96).

We compute ( I I .95) from to BP *RP:: , we can work modulo

(I 1 .99)

(4)

(21)

r 1 ( y4 ) = a0 x 1 1

+ a1 x 1 2

(1 1 .98). + a2 x 1 3 '

(I I .l OO) from (I I .8 8) (I l .IOI) e(- I 7v21 + 68v 1 b 1 - 68b 21 ) - - I7v21 1 because e(b) = I . Since 2x 3 = 0, (I l .102) Computing r2 (y 4 ) = 2z 2 is about the same . So (I 1.103) r(y4) = 4z 1 + 2z 2 • Note that for. our purposes we need only the table of e o r from (I I .88), as *

79

BROWN-PETERSON HOMOLOGY €

deg

0

'*

Fo --+-

coefficients of 1u

mod

113

b3 - o 11

b 31 --+- 1 lu

13

( I 1 . 1 04)

--+- - 2

b1 -+- 2v 1

-1

b 1 b 3 --+- - v t

2

b l l l l --+- - v 1

2

b 15 - o

0

b 21 b 3 -+ v21

0

b f 1 1 1 --+- 0

0

15

A similar

(8)

calculation to that for

r( y 4 )

(4)

(2)

also gives

{ 1 1 . 1 05) Computing ( 1 1 . 1 06)

r(y2) is

y2 = v2x 1 1 . We have (r(y2), Pk> = e o r.(v2x 1 1 ).( Pk ) = e o r.v2 .x V ( Pk) slightly different because

where v2 ,. is j ust multiplication by 17R (v2). Recall that in ( 1 1 .1 04) there is an absent 1 5 , and bi going to zero. We compute

r(y2) = 0 .

(1 1 . 1 07)

Computing r(y 1 ) is fairly easy. We represent it as ( 1 1 . 1 08)

3 l2 r( y t ) = aoxs + at x9 + a2x t o + a 3x 1 1 + a4x + a s x t .

Then

ao at a2 ( 1 1 . 1 09)

03

=eo

r. x!

Ps

?

0

0

p9

?

0

0

0

0

0

0

0

2

Pt o Pu

=

€

0

'•

? 0

04

(:J1 2

2b 41

as

Pu

0

=

0 0

bt >

80 and we have ( l l . l lO)

W. STEPHEN WILSON

We are almost done. We want to show the lack of surjectivity of (1 1 .83), and it is now clear, from (1 1.103), (l l .lO S), (1 1 .107) and (1 1 .1 10), put into (1 1 .95), that y 1 cannot be in the image in (1 1 .83). We conclude this section with some formulas we have found useful. We use the formu· las and tables found earlier in the paper, and 1 THEOREM 1 1 .1 1 1 . Let [p) (x) = �j;oaix i + , with b �i � O bi = c(�i� O t;), and a = �i� O a i , ai E BP2 i . Then, with x(y) = �i � O xi y i + I , b(a ) 11R a(b ), or 1 L b; a i + = L (a; i 1 . =

i� O

i� O

TIR

)b +

=

o

PRooF. Just stabilize 9. 1 2 for BP. o This allows us to compute the right unit of v's in terms of b's and give the nongen erating b's in terms of the generating 1 b's. p = 2, [2] (x) = �i � O ai xi + • In BP*BP

2, 11R (a i ) = -v l + 2b t , TJR (a 2 ) = 2vi - 8v 1 b 1 + 8b i , TJR (a 3 ) = - 7v2 - 8v : + 13vf b 1 + 9v 1 b f + 14b3 - 34b : , 11R (a 4 ) = 3 0v1 v2 + 26vt - 60v2b 1 - 58v: b1 - 126vi b i (1 1.1 12) - 60v 1 b3 + 488v1 b � - 424b i + 120b 1 b3, TJR (a5) = - 1 l lv2vi - 84v � + 444v2v 1 b 1 + 2 8 Svi b 1 - 444v2 b i + 525_v � b f + 222vi b3 - 3714vi b � - 888v 1 b 1 b3 + 6156v 1 b i + 888b i b3 - 3528b � , 11R (a 6 ) = 1 1 2vi + 502v2v � + 300v� - 1892vf v2b1 - 1090v � b 1 + 2664v1 v2b f - 1 790vib f - 448v2b3 - 1004v� b3 - 880v2 b � + 1 7528v� b � + 3784vf b3b 1 - 39 728vf b i - 5328v 1 b i b3 + 39936v1 b � + 448b i + 1760b3b : - 1 5072b �. 11R (a0 ) =

81

BROWN PETERSON HOMOLOGY

= 2, '1R (2) = 2, '1R (v1 ) = v1 - 2b 1 b2 = -v1 b1 + 2b�, '1R (v2 ) = v2 + 5v�b1 - 15v1b� - 2b3 + 14b:, b4 = -v:b1 - 2v2 b1 - 3v�b� + 14v1 b: - v1b3 - 1 6b1 + 6b1b3, b 5 = 4v2 v1b1 + 2v1b1 - l lv2 b� + v:b� + v�b 3 - 49v�b: - 13v1b1b3 + 121v1 b1 + 28b�b3 - 98b�, b 6 = -4v�v2 b1 - 2v�b1 + 27v1 v2 b � + 7v1b � - 2v2 b3 - 2v:b 3 - 40v2 b: + S7v:b: + 13v�b3b1 - 3 2 1 v�b 1 - 60v 1 b�b 3 + 570v1b� + 4b� + 80b:b3 - 368b� .

In BP.BP, p

•

(1 1 . 1 1 3)

a0 = 1 , a 1 = - 1 + v 1 - v� + (2v2 + 2v:) - (3v1 + 4v1 v2 ) + (6v�v2 + 4v: ) + · · · , a2 = 2 - Sv1 + Bv� - (17v: + 1 1v2 ) + (37v1 v2 + 34v1) + · · · , a 3 = - s + 2lv 1 - 49v21 + (SOv2 + 1 18v31 ) + · ' a0 1 - - 1 + v 1 - v21 + (2v2 + 2v 13 ) + . , a4 = 14 - 84v1 + 264v� + . . · , a 1 1 = 6 - 13v1 + 2lv� + . · , a 5 = -42 + 330v1 + · · , a2 1 = - 28 + 100v1 + . . . , a 6 = 132 + . . . , a 3 1 = 1 20 + . . . , ao 2 = 4 + . . . . . .

.

( 1 1 . 1 1 4)

.

·

·

·

References

[A 1 ] J. F. Adams, Lectures on generalized cohomology, Lecture Notes in Math., vol. 99, Springer-Verlag, Berlin, 1969, pp. 1-138. [A2 ] --, Quillen 's work on formal groups and complex cobordism, Stable Homo topy and Generalized Homology, Univ. of Chicago Press, Chicago, Ill., 1974, pp. 29-1 20. [A 3 ] -- , On the groups J(X). IV, Top ol ogy S (1966), 21-71. [A4] -- , Localization and completion with an addendum on the use of Brown Peterson homology in stable homotopy, Univ. of Chicago Lecture Notes in Mathematics, 1 975.

[Ar 1 ] S. Araki, Typical formal groups in complex cobordism and K-theory, Lectures Dept. of Math., Kyoto Univ., Tokyo, Kinokuniya Book Store, 1973. [Ar 2 ] -- , Multip li ative operatiom in BP cohomology, Osaka J. Math. 1 2 (1975),

in Math.,

343-3 56.

c

N. A. Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 ( 1 973), 279-302. [BCM] M. Bendersky, E. B. Curtis and H. R. Miller, The unstable Adams spectral sequence for generalized homology, Topology 1 7 (1978), 229-248. [Br] E. H. Brown, Cohomology theories, Ann. of Math. (2) 75 (1962), 467-484. [BP] E. H. Brown and F. P. Peterson, A spectrum whose ZP -cohomology is the alge bra of reduced p-th po wers, Topology S ( 1 9 6 6) , 1 49 - 1 5 4 . [Ch] K. Chan, Applications of the bar and cobar spectral sequences to the Brown Peterson spectrum, Thesis , The Johns Hopkins University , 1980. [C) P. E. Conner, Differentiable periodic maps, 2nd ed., Graduate Texts in Math., no. 738, Springer-Verlag, Berlin, 1 978. [CF 1 ] P. E. Conner and E. E. Floyd, Differentill periodic maps, Springer-Verlag, Berlin, 1964. (CF2 ] -- , The relation of cobordism to K-theories, Lecture Notes in Math., vol. 28, Springer-Verlag, Berlin, 1966. [CF 3 ] -- , Torsion in SU-bordism, Mem . Amer. Math. Soc., no. 60, Amer. Math. Soc., Providence, R. 1 . , 1966. [CS] P. E. Conner and L. Smith, On the complex bordism of finite complexes, Inst. Hautes Etudes Sci. Publ . Math. 37 (1969), 1 17-221 . [tD] T. tomDieck, Actions of finite abelian p-groups without stationary points, To pology 9 (1970), 359-366. 83

[Ba]

84

W. STEPHEN WILSON

[F) E. E. Floyd, Actions of (Zpl without stationary points, Topology 1 0 (1971), 327-336. [G) V. Giambalvo, Some tables for formal groups and BP, Geometric Applications of Homotopy Theory. II, Lecture Notes in Math., vol. 658, Springer-Verlag, Berlin, 1978. [H1 ] M. Hazewinkel, A universal formal group and complex cobordism, Bull. Amer. Math. Soc. 8 1 (1975), 930-933. [H2 ] -- , Constructing formal groups. I I I . Applications to complex cobordism and Brown-Peterson cohomology, J. Pu re Appl. Algebra 1 0 (1977/78), 1-18. (JMWZ] D. C. Johnson, H. R. Miller, W. S. Wilson and R. S. Zahler, Boundary homo morphisms in the generalized Adams spectral sequence and the non-triviality of infinitely many "ft in stable homotopy, Reunion Sobre Teoria de Homotopia (Donald Davis, ed.),

Notas de Matematicas y Simposia, no. 1 , Sociedad Matematicas Mexicana, Mexico City, 1975, pp. 47-59. [JW. J D. C. Johnson and W. S. Wilson, Projective dimension and Brown-Peterson homology, Topology 1 2 (1973), 327-353. [JW2 ] -- , BP operations and Morava 's extraordinary K-theories, Math. Z. 1 44 ( 1 9 7 5), 55-75. [JY) D; C. Johnson and Z . Yosimura, Torsion in Brown-Peterson homology and Hure wicz homomorphisms, Osaka J. Math. 1 7 (1980), 1 17- 1 36. [K] I . Kozma, Witt vectors and complex cobordism, Topology 1 3 (1974), 389;_394. [L1 ] P. S. Landweber, Cobordism operations and Hop[ algebras, Trans. Amer. Math. Soc. 1 29 (1967), 94- 1 10. , Annihilator ideals and primitive elements in complex bordism, Illinois J. [ L2 ] Math. 1 7 (1973), 273-284. [L 3 ] , Associated prime ideals and Hop[ algebra, J. Pure A p pl . Algebra 3 (1973), 43-58. [L4] , Homological properties of comodules over MU*MU and BP*BP, Amer. J. Math. 98 (1976), 591-610. [L5] , New applications of commutative algebra to Brown-Peterson homology, Lecture Notes in Math., vol. 741 , Springer-Verlag, Berlin, 1979, pp. 449-460. [Li] A. Liulevicius, On the algebra BP* (BP), Lecture Notes in Math., vol. 249, Springer Verlag, Berlin, 1 97 1 , pp. 47-53. [ML] S. MacLane, Categories for the working mathematician, Springer, Berlin, 1 97 1 . [Ma] J . Martin, A n algorithm which generates basis elements for the homology of the Brown-Peterson spectrum, Thesis, The Johns Hopkins University, 1 98 1 . [Mg] R. J. Milgram, The bar construction and abelian H-spaces, Ilinois J . Math. --

--

--

--

ll (1 967), 242-250.

[M] H. R. Miller, Some algebraic aspects of the Adams-Novikov spectral sequence, Thesis, Princeton University, 1 974. [MR] H. R. Miller and D. C. Ravenel, Morava stabilizer algebras and the localization 's E2-term, Duke Math. J. 44 (1977), 433-447. Novikov of

BROWN PETE RSON HOMOLOGY

[M RW]

H.

R. Mi ll e r ,

D. C.

Adams-Novikov spectral sequence, [MW]

H.

Ravenel and W. S. Wilson , Periodic phenomena in the A n n . o f M ath.

(2) I 06 ( 1 977), 469--5 1 6 .

R. Mille r and W. S . Wilson , On Novikov 's Ext 1 modulo an invan"ant prime

To p ol ogy 1 5 { 1 976), 1 3 1 - 1 4 1 . [Mi 1 ] J . W . Mil nor , On the cobordism ring n * and a complex analogue. I , A mer . M a th . 82 ( 1 960), 505-5 2 1 . [ M i2 ] -- , The Steenrod algebra and its dual, Ann. of M a th . (2} 67 ( 1 958), 1 50- 1 7 1 . ideal,

J.

[Mo 1 ]

J.

Morava, Structure theorems for cobordism comodules, p repri n t .

[ Mo2] -- , Extensions of cobordism comodules,

pre print .

-- , A product for odd-primary bordism of manifolds with singularities,

[M o 3 )

Topology 1 8 { 1 979), 1 77- 1 86 . lN)

burdism

S . P. Novik ov, The methods of algebraic topology from the view point of co

theories, M a th .

31 ( 1 967), 855 -9 5 1 .

lOT)

S . Oka a n d

Hi r osh ima

USSR-Izv. I

H.

( 1 967), 82 7-9 1 3 = l zv . Aka d . Nauk SSSR Ser. Mat.

T o da, Nontriviality of an element in the stable homotopy groups

Math . J . 5 { 1 975). 1 1 5- 1 25 . Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Am e r . M a th . Soc. 7 5 ( 1 969), 1 293- 1 298 . [ Q2 ] -- , Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. in Math. 7 ( 1 97 1 ), 29--56.

of spheres,

[Q1 )

D.

[R1 ]

D. C . Ravenel , The structure of BP * BP modulo a n invarian t prime ideal, Topol

[ R2 ]

---- , The structure of Morava stabilizer algebras, I nve n t . M a th . 37

ogy 1 5 { 1 976), 1 49- 1 5 3 . 1 09- 1 20.

-- , Localization with respect to certain periodic homology theories, pre

[R3 ) p rin t .

D. C . Rave nel a n d W . S .

( RW 1 ) Pure Appl .

Algebra

[RW2 )

9

[ RS ]

Bull .

[SY ]

Wil son ,

M.

J.

Math . 1 02 ( 1 980), 69 1 -748 .

Rothe nbe rg and N . Stee nro d , The cohomology of classifying spaces of If

A me r . Math . Soc .

N . Shimada an d

7 1 { 1 965), 8 72-8 7 5 ; mimeographed notes, Prince ton Univ.

N. Yagita, Multiplications in complex bordism theory with

K.

{ 1 976), 259-293 .

Sinkinson , The cohomology of certain spectra associated with the Brown

Peterson spectrum, Duke Math . J . 43 [Sm 1 ]

J.

-- , The Morava K-theories of Eilenberg-MacLane spaces and the Conner

singularities, Pu bl . Res . I ns t . Math . Sci . 1 2 [Si ]

The Hop[ ring for complex cobordism,

{ 1 977), 24 1 -280.

Floyd conjecture, Amc r . spaces,

( 1 9 7 6),

L.

{ 1 976 ), 605-622 .

Smith, On the complex bordism offinite complexes, Proc. A d van ce d Study

Various 5 1 3-566.

Institute on Algebraic Topology , vol . I I I , stitute , Aarh us Unive rsitet,

1 9 70,

pp.

Publication Series No .

13,

Matema tisk I n

86

W STEPHEN WILSON [Sm2 ]

793-856. (Sn]

--

, On realizing complex bordism modules, Amer. J. Math.

V. P. Snaith,

No. 221 (1979).

92

(1970),

Algebraic cobordism and K-theory, Mem. Amer. Math. Soc.

(St] R. Stong, Notes on cobordism theory, Math. Notes, Princeton Univ. Press, Princeton, N. J., [Su] D. Sullivan, Singularities in spaces, Proc. Liverpool Singularities Sympos. II, Lec ture Notes in Math., vol Springer-Verlag, Berlin, pp . [Sw] R. M . Switzer, Algebraic topology-homotopy and homology, Springer, Berlin,

1968.

209,

197 1 ,

196-207.

1975.

[T) R. Thorn, Quelques proprietes globales des varietes differentiables, Comment. Math. Helv. 28 [TZ 1 ) E. Thomas and R. S. Zahler, Nontriviality of the stable homotopy element r 1 , J. Pure Appl. Algebra 4 [TZ 2 ] Generalized higher order cohomology operations and stable homotopy groups of spheres, Adv. in Math. 20 [TW) R. W. Thomason and W. S. Wilson, Hopf rings in the bar spectral sequence, Quart. J. Math. 3 1 (To 1 ) H. Toda , On realizing exterior parts of the Steenrod algebra, Topology

(1954), 17-86.

--

,

(1974), 1 89-203.

(1976), 289-328.

(1980), 507-5 1 1 .

10

(1971), 53-65.

, (1959), 297-332.

(To 2 ] p-pn"mary components of homotopy groups. IV, Mem. Coli. Sci. Kyoto Univ. Ser. A 32 [V] R . Vogt, Boardman 's stable homotopy category, Lecture Notes Series Mate· matisk lnstitut, Aarhus Universitet, [Wh) G. W. Whitehead , Generalized homology theories, Trans. Amer. Math. Soc. --

21,

1970.

1 02

(1962), 227-283.

[W 1 ) W . S. Wilson, The il-spectrum for Brown-Peterson cohomology. l , Comment. Math. Helv. 48 (W2 ] --, The il-spectrum for Brown-Peterson cohomology. II , Amer. J. Math.

(1973), 45-55.

97

(1975), 101-123.

On products in a family of cohomology theories associated to the invariant prime ideals of 1T* (BP), Comment. Math. Helv. 52 (1977), 457-481 . [Wu 2 ] -- , On the relation of Morava K-theories to Brown-Peterson homology, (Wu t l U . Wiirgler,

Topology and Algebra, Proc. Colloq in Honor of B. Eckmann (Zurich pp.

269-280.

1977), 1978,

A splitting theorem for certain cohomology theories associated to BP* (- ), Manuscripta Math. 29 (Zb] A. Zabrodsky, Hopf spaces, North-Holland Math. Studies, vol. Notas de Matematica, no. North-Holland, Amsterdam, (Zh] R. S. Zahler, Fringe families in stable homotopy of spheres, Trans. Amer. Math. Soc. 224

(1979), 93-1 1 1 . 59,

(1976), 243-253.

1976.

22;

Brown-Peterson Homology: An Introduction and Sampler (Cbms Regional Conference Series in Mathematics 48)

Lectures on Three-Manifold Topology (Regional conference series in mathematics) (Cbms Regional Conference Series in Mathematics)

Topological Quantum Computation (Cbms Regional Conference Series in Mathematics)