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Home , Gunnar Carlsson, Homotopy, Homotopy theory

EQUIVARIANT AND

THEORY

May JP

The author acknowledges the supp ort of the NSF

Subject Classication Revision Primary L M

N N N N N P P P P Q R

R T R Secondary E G G A P P

P Q Q R R S S S U U R

R S

Author addresses

University of Chicago Chicago Il

Email address maymathuchicagoedu ii

Contents

Intro duction

Chapter I Equivariant Cellular and Theory

Some basic denitions and adjunctions

Analogs for based Gspaces

GCW complexes

Ordinary homology and cohomology theories

ersal co ecient sp ectral Univ

Chapter I I P ostnikov Systems Lo calization and Completion

Gspaces and Postnikov systems Eilenb ergMacLane

lo calizations of spaces Summary

Lo calizations of Gspaces

Summary completions of spaces

Completions of Gspaces

Chapter I I I Equivariant Rational

Summary the theory of minimal mo dels

Equivariant minimal mo dels

Rational equivariant Hopf spaces

Chapter IV Smith Theory

cohomology Smith theory via Bredon iii

iv CONTENTS

Borel cohomology lo calization and Smith theory

Equivariant Applications Chapter V Categorical Constructions

Co ends and geometric realization

Homotopy colimits and limits

Elmendorf s theorem on diagrams of xed p oint spaces

spaces Eilenb ergMac Lane Gspaces and universal F

Chapter VI The Homotopy Theory of Diagrams

Elementary homotopy theory of diagrams

Homotopy Groups

Cellular Theory

The homology and cohomology theory of diagrams

J

The closed mo del structure on U

Another pro of of Elmendorf s theorem

Chapter VI I Equivariant theory and Classifying Spaces

The denition of equivariant bundles

The classication of equivariant bundles

Some examples of classifying spaces

Chapter VI I I The Sullivan

Statements of versions of the Sullivan conjecture

and Fix Algebraic preliminaries Lannes T

Lannes generalization of the Sullivan conjecture

Sketch pro of of Lannes theorem

Maps b etw een classifying spaces

An intro duction to equivariant stable homotopy Chapter IX

Gspheres in homotopy theory

GUniverses and stable Gmaps

and transfer Gmaps

Mackey functors and coMackey functors

Ggraded homology and cohomology RO

CONTENTS v

The Conner conjecture

V complexes and RO Ggraded cohomology Chapter X GCW

Motivation for cellular theories based on representations

GCWV complexes

Homotopy theory of GCWV complexes

Ordinary RO Ggraded homology and cohomology

Chapter XI The equivariant Hurewicz and Susp ension Theorems

Background on the classical theorems

Formulation of the problem and counterexamples

An oversimplied description of the results

The statements of the theorems

Sketch pro ofs of the theorems

Chapter XI I The Equivariant Stable Homotopy

An intro ductory overview

Presp ectra and sp ectra

Smash pro ducts

Function sp ectra

The equivariant case

Spheres and homotopy groups

GCW sp ectra

Stability of the stable category

Getting into the stable category

Chapter XI I I RO Ggraded homology and cohomology theories

for ROGgraded cohomology theories

Representing RO Ggraded theories by Gsp ectra

Browns theorem and RO Ggraded cohomology

Equivariant Eilenb ergMacLane sp ectra

Gsp ectra and pro ducts

theory Chapter XIV An intro duction to equivariant K

vi CONTENTS

The denition and basic prop erties of K theory

G

Bundles over a p oint the representation ring

Equivariant Bott p erio dicity

Equivariant K theory sp ectra

The AtiyahSegal completion theorem

The generalization to families

Chapter XV An intro duction to equivariant cob ordism

A review of nonequivariant cob ordism

Equivariant cob ordism and Thom sp ectra

Computations the use of families

Sp ecial cases o dd order groups and Z

Chapter XVI Sp ectra and Gsp ectra change of groups duality

Fixed p oint sp ectra and orbit sp ectra

Split Gsp ectra and free Gsp ectra

Geometric xed p oint sp ectra

uller isomorphism Change of groups and the Wirthm

Quotien t groups and the Adams isomorphism

sp ectra from Gsp ectra The construction of GN

SpanierWhitehead duality

V duality of Gspaces and Atiyah duality

Poincar e duality

Chapter XVI I The Burnside ring

Generalized Euler characteristics and transfer maps

G

The Burnside ring AG and the zero stem S

0

Prime ideals of the Burnside ring

Idemp otent elements of the Burnside ring

Lo calizations of the Burnside ring

Lo calization of equivariant homology and cohomology

Chapter XVI I I Transfer maps in equivariant bundle theory

CONTENTS vii

The transfer and a shifting variant

Basic prop erties of transfer maps

Smash pro ducts and Euler characteristics

The double coset formula and its applications

Transitivity of the transfer

Chapter XIX Stable homotopy and Mackey functors

The splitting of equivariant stable homotopy groups

Generalizations of the splitting theorems

Equivalent denitions of Mackey functors

Induction theorems

Splittings of rational Gsp ectra for nite groups G

Chapter XX The Segal conjecture

The statement in terms of completions of Gsp ectra

A calculational reformulation

groups A generalization and the reduction to p

The pro of of the Segal conjecture for nite pgroups

Approximations of singular subspaces of Gspaces

An inverse limit of Adams sp ectral sequences

Further generalizations maps b etween classifying spaces

Chapter XXI Generalized Tate cohomology

Denitions and basic prop erties

Ordinary theories AtiyahHirzebruch sp ectral sequences

Cohomotopy p erio dicity and ro ot invariants

The generalization to families

Equivariant K theory

Further calculations and applications

Chapter XXI I Brave new algebra

The category of S mo dules

Categories of Rmo dules

viii CONTENTS

The algebraic theory of Rmo dules

The homotopical theory of Rmo dules

Categories of Ralgebras

Bouseld lo calizations of Rmo dules and algebras

Top ological Ho chschild homology and cohomology

Chapter XXI I I Brave new equivariant foundations

Twisted halfsmash pro ducts

sp ectra The category of L

A ring sp ectra and S algebras and E

1 1

Alternative p ersp ectives on equivariance

The construction of equivariant algebras and mo dules

Comparisons of categories of LGsp ectra

Equivariant Algebra Chapter XXIV Brave New

Intro duction

Cech cohomology in algebra Lo cal and

Brave new versions of lo cal and Cech cohomology

Lo calization theorems in equivariant homology

Completions completion theorems and lo cal homology

A pro of and generalization of the lo calization theorem

The application to K theory

Lo cal Tate cohomology

Chapter XXV Lo calization and completion in complex b ordism

The lo calization theorem for stable complex b ordism

An outline of the pro of

The norm and its prop erties

The idea b ehind the construction of norm maps

functors with smash pro duct Global I



The denition of the norm map

The splitting of M U as an algebra G

CONTENTS ix

Loers completion conjecture

Chapter XXVI Some calculations in complex equivariant b ordism

Notations and terminology

Stably almost complex structures and b ordism

Tangential structures

Calculational to ols

Statements of the main results

1

S Preliminary lemmas and families in G

1

On the families F in G S

i

1

Passing from G to G S and G Z

k

Bibliography

Intro duction

This volume b egan with Bob Piacenzas suggestion that I b e the principal lecturer

Regional Conference in Fairbanks Alaska That event to ok at an NSFCBMS

place in August of and the interim has seen very substantial progress in this

general area of mathematics The scop e of this volume has grown accordingly

ariant algebraic top ology to sta The original fo cus was an intro duction to equiv

ble homotopy theory and to equivariant that was geared

towards graduate students with a reasonably go o d understanding of nonequivari

ant algebraic top ology More recent material is changing the direction of the last

by allowing the intro duction of p ointset top ological algebra into sta two sub jects

ble homotopy theory b oth equivariant and nonequivariant and the last p ortion

of the b o ok fo cuses on an intro duction to these new developments There is a

progression with the later p ortions of the b o ok on the whole b eing more dicult

than the earlier p ortions

Equivariant algebraic top ology concerns the study of algebraic invariants of

two chapters intro duce the basic structural spaces with actions The rst

foundations of the sub ject cellular theory ordinary homology and cohomology

theory Eilenb ergMac Lane Gspaces Postnikov systems lo calizations of Gspaces

and completions of Gspaces In most of this work G can b e any top ological group

but we restrict attention to compact Lie groups in the rest of the b o ok

Chapter I I I on equivariant was written by Georgia

Triantallou In it she shows how to generalize Sullivans theory of minimal

mo dels to obtain an algebraization of the of nilp otent G

spaces for a nite group G This chapter contains a rst surprise rational Hopf

Gspaces need not split as pro ducts of Eilenb ergMac Lane Gspaces This is a hint

INTRODUCTION

that the calculational b ehavior of equivariant algebraic top ology is more intricate

and dicult to determine than that of the classical nonequivariant theory

gives two pro ofs of the rst main theorem of equivariant algebraic Chapter IV

theory any xed p oint space top ology which go es under the name of Smith

of an action of a nite pgroup on a mo d p homology sphere is again a mo d p

homology sphere One pro of uses ordinary or Bredon

and the other uses a general lo calization theorem in classical or Borel equivariant

cohomology

Parts of equivariant theory require a go o d deal of categorical b o okkeeping for

example to keep track of xed p oint data and to construct new Gspaces from

diagrams of p otential xed p oint spaces Some of the relevant background such

as geometric realization of simplicial spaces and the construction of homotopy

colimits is central to all of algebraic top ology These matters are dealt with

in Chapter V where Eilenb ergMac Lane Gspaces and universal F spaces for

families F of subgroups of a given group G are constructed Sp ecial cases of

such universal F spaces are used in Chapter VI I to study the classication of

equivariant bundles

t p ersp ective on these matters is given in Chapter VI which was writ A dieren

ten by Bob Piacenza It deals with the general theory of diagrams of top ological

spaces showing how to mimic classical homotopy and homology theory in cat

egories of diagrams of top ological spaces In particular Piacenza constructs a

Quillen closed mo del category structure on any such category of diagrams and

shows how these ideas lead to another way of passing from diagrams of xed p oint

spaces to their homotopical realization by Gspaces

Chapter VI I I combines equivariant ideas with the use of new to ols in nonequiv

ariant algebraic top ology notably Lannes T in the context of unstable

mo dules and algebras over the Steenro d algebra to describ e one of the most b eau

tiful recent developments in algebraic top ology namely the Sullivan conjecture

many mathematicians have contributed to this area and its applications While

the main theorems are due to Gunnar Carlsson and Jean Lannes

X Y of homotopy classes of based maps from a space X to a Although the set

space Y is trivial to dene it is usually enormously dicult to compute The Sulli

van conjecture in its simplest form asserts that B G X if G is a nite group

and X is a nite CW complex It admits substantial generalizations which lead

to much more interesting calculations for example of the set of maps BG BH

INTRODUCTION

for suitable compact Lie groups G and H We shall see that an understanding

of equivariant classifying spaces sheds light on what these calculations are really

saying There is already a large literature in this area and we can only give an in

tro duction One theme is that the Sullivan conjecture can b e viewed conceptually

as a calculational elab oration of Smith theory A starting p oint of this approach

lies in work of Bill Dwyer and Clarence Wilkerson which rst exploited the study

of mo dules over the Steenro d algebra in the context of the lo calization theorem in

Smith theory

We b egin the study of equivariant stable homotopy theory in Chapter IX which

gives a brief intro duction of some of the main ideas The chapter culminates with

a quick conceptual pro of of a conjecture of Conner if G is a compact Lie group

and X is a nite dimensional GCW complex with nitely many orbit typ es such

that H X Z then H XG Z This concrete statement is a direct con

of the seemingly esoteric assertion that ordinary equivariant cohomology

with co ecients in a Mackey functor extends to a cohomology theory graded on

the real representation ring RO G this means that there are susp ension isomor

phisms with resp ect to the based spheres asso ciated to all representations not just

trivial ones In fact the interplay b etween homotopy theory and representation

theory p ervades equivariant stable homotopy theory

One manifestation of this app ears in Chapter X which was written by Stefan

Waner It explains a variant theory of GCW complexes dened in terms of repre

sentations and uses the theory to construct the required ordinary RO Ggraded

cohomology theories with co ecients in Mackey functors by means of appropriate

cellular co chain complexes

Another manifestation app ears in Chapter XI which was written by Gaunce

Lewis and which explains equivariant versions of the Hurewicz and Freudenthal

susp ension theorems The algebraic transition from unstable to stable phenom

ena is gradual rather than all at once Nonequivariantly the homotopy groups

of rst lo op spaces are already Ab elian groups as are stable homotopy groups

Equivariantly stable homotopy groups are mo dules over the Burnside ring but

the homotopy groups of V th lo op spaces for a representation V are only mo d

ules over a partial Burnside ring determined by V The precise form of Lewiss

equivariant susp ension theorem reects this algebraic fact

Serious work in b oth equivariant and nonequivariant stable homotopy theory

requires a go o d category of stable spaces called sp ectra in which to work

INTRODUCTION

There is a great deal of literature on this sub ject The original construction of the

nonequivariant stable homotopy category was due to Mike Boardman One must

make a sharp distinction b etween the stable homotopy category which is xed

and unique up to equivalence and any particular p ointset level construction of

it In fact there are quite a few constructions in the literature However only one

of them is known to generalize to the equivariant context and that is also the one

that is the basis for the new development of p ointset top ological algebra in stable

homotopy theory We give an intuitive intro duction to this category in Chapter

XI I b eginning nonequivariantly and fo cusing on the construction of smash pro d

ucts and function sp ectra since that is the main technical issue We switch to

the equivariant case to explain homotopy groups the susp ension isomorphism for

representation spheres and the theory of GCW sp ectra We also explain how to

transform the sp ectra that o ccur in nature to the idealized sp ectra that are the

ob jects of the stable homotopy category

In Chapters XI I I XIV and XV we intro duce the most imp ortant RO G

graded cohomology theories and describ e the Gsp ectra that represent them We

b egin with an axiomatic account of exactly what RO Ggraded homology and

cohomology theories are and a pro of that all such theories are representable by G

sp ectra We also discuss ring Gsp ectra and pro ducts in homology and cohomology

theories We show how to construct Eilenb ergMac Lane Gsp ectra by representing

the zeroth term of a Zgraded cohomology theory dened by means of Gsp ectrum

level co chains This implies an alternative construction of ordinary RO Ggraded

cohomology theories with co ecients in Mackey functors

Chapter XIV which was written by John Greenlees gives an intro duction to

equivariant K theory The fo cus is on equivariant Bott p erio dicity and its use to

prove the AtiyahSegal completion theorem That theorem states that for any

compact Lie group G the nonequivariant K theory of the BG is

isomorphic to the completion of the representation ring RG at its augmentation

ideal I The result is of considerable imp ortance in the applications of K theory

and it is the prototyp e for a numb er of analogous results to b e describ ed later

Chapter XV which was written by Steve Costenoble gives an intro duction to

equivariant cob ordism The essential new feature is that transversality fails in gen

eral so that geometric equivariant b ordism is not same as stable or homotopical

b ordism the latter is the theory represented by the most natural equivariant gen

eralization of the nonequivariant Thom sp ectrum Costenoble also explains the

INTRODUCTION

use of adjacent families of subgroups to reduce the calculation of equivariant b or

dism to suitably related nonequivariant calculations The equivariant results are

considerably more intricate than the nonequivariant ones While the Gsp ectra

that represent unoriented geometric b ordism and its stable analog split as pro d

ucts of Eilenb erg Mac Lane Gsp ectra for nite groups of o dd order just as in the

nonequivariant case this is false for the cyclic group of order

Chapters XVIXIX describ e the basic machinery and results on which all work

in equivariant stable homotopy theory dep ends Chapter XVI describ es xed p oint

and orbit sp ectra shows how to relate equivariant and nonequivariant homology

and cohomology theories and more generally shows how to relate homology and

cohomology theories dened for a group G to homology and cohomology theories

dened for subgroups and quotient groups of G These results ab out change of

groups are closely related to duality theory and we give basic information ab out

equivariant SpanierWhitehead Atiyah and Poincar e duality

In Chapter XVI I we discuss the Burnside ring AG When G is nite AG is

the Grothendieck ring asso ciated to the semiring of nite Gsets For any compact

Lie group G AG is isomorphic to the zeroth equivariant stable

G G

X X of spheres It therefore acts on the equivariant homotopy groups

n

n

of any Gsp ectrum X and this implies that it acts on all homology and cohomology

groups of any Gsp ectrum Information ab out the algebraic structure of AG

leads to information ab out the entire stable homotopy category of Gsp ectra It

turns out that AG has Krull dimension one and an easily analyzed prime ideal

sp ectrum making it quite a tractable ring Algebraic analysis of lo calizations of

AG leads to analysis of lo calizations of equivariant homology and cohomology

theories For example for a nite group G the lo calization of any theory at a

prime p can b e calculated in terms of sub quotient pgroups of G

In Chapter XVI I I we construct transfer maps which are basic calculational

to ols in equivariant and nonequivariant bundle theory and describ e their basic

prop erties Sp ecial cases were vital to the earlier discussion of change of groups

The deep est prop erty is the double coset formula and we say a little ab out its

applications to the study of the cohomology of classifying spaces

In Chapter XIX we discuss several fundamental splitting theorems in equiv

ariant stable homotopy theory These describ e the equivariant stable homotopy

groups of Gspaces in terms of nonequivariant homotopy groups of xed p oint

spaces These theorems lead to an analysis of the structure of the sub category

INTRODUCTION

of the stable category whose ob jects are the susp ension sp ectra of orbit spaces

A Mackey functor is an additive contravariant functor from this sub category to

Ab elian groups and when G is nite the analysis leads to a pro of that this

top ological denition of Mackey functors is equivalent to an earlier and simpler al

gebraic denition Mackey functors describ e the algebraic structure that is present

H H

on the system of homotopy groups X X of a Gsp ectrum X where

n

n

G

X is H runs over the subgroups of G The action of the Burnside ring on

n

part of this structure It is often more natural to study such systems than to fo cus

on the individual groups In particular we describ e algebraic induction theorems

that often allow one to calculate the value of a Mackey functor on the orbit GG

from its values on the orbits GH for certain subgroups H Such theorems have

applications in various branches of mathematics in which nite group actions ap

p ear Again algebraic analysis of rational Mackey functors shows that when G is

nite rational Gsp ectra split as pro ducts of Eilenb ergMac Lane Gsp ectra This

is false for general compact Lie groups G

In Chapter XX we turn to another of the most b eautiful recent developments

in algebraic top ology the Segal conjecture and its applications The Segal con

jecture can b e viewed either as a stable analogue of the Sullivan conjecture or

as the analogue in equivariant stable cohomotopy of the AtiyahSegal completion

theorem in equivariant K theory The original conjecture which is just a fragment

of the full result asserts that for a nite group G the zeroth stable cohomotopy

group of the classifying space BG is isomorphic to the completion of AG at its

augmentation ideal I The key step in the pro of of the Segal conjecture is due

to Gunnar Carlsson We explain the pro of and also explain a numb er of general

izations of the result One of these leads to a complete algebraic determination

of the group of homotopy classes of stable maps b etween the classifying spaces

of any two nite groups This is analogous to the role of the Sullivan conjecture

in the study of ordinary homotopy classes of maps b etween classifying spaces

Use of equivariant classifying spaces is much more essential here In fact the Se

gal conjecture is intrinsically a result ab out the I adic completion of the sphere

Gsp ectrum and the application to maps b etween classifying spaces dep ends on

a generalization in which the sphere Gsp ectrum is replaced by the susp ension

Gsp ectra of equivariant classifying spaces

Chapter XXI is an exp osition of joint work of John Greenlees and myself in which

we generalize the classical Tate cohomology of nite groups and the p erio dic cyclic

INTRODUCTION

cohomology of the circle group to obtain a Tate cohomology theory asso ciated to

any given cohomology theory on Gsp ectra for any compact Lie group G This

work has had a variety of applications most strikingly to the computation of the

top ological cyclic homology and thus to the algebraic K theory of numb er rings

While we shall not get into that application here we shall describ e the general

AtiyahHirzebruchTate sp ectral sequences that are used in that work and we shall

give a numb er of other applications and calculations For example we shall explain

a complete calculation of the Tate theory asso ciated to the equivariant K theory

of any nite group This is an active area of research and some of what we say

at the of this chapter is rather sp eculative The Tate theory provides some of

the most striking examples of equivariant phenomena illuminating nonequivariant

phenomena and it leads to interrelationships b etween the stable homotopy groups

of spheres and the Tate cohomology of nite groups that have only b egun to b e

explored

Chapters XXI I through XXV concern brave new algebra the study of p oint

set level top ological algebra in stable homotopy theory The desirability of such

a theory was advertised by Waldhausen under the rubric of brave new rings

hence the term brave new algebra for the new sub ject Its starting p oint is the

construction of a new category of sp ectra the category of S mo dules that has a

smash pro duct that is symmetric monoidal asso ciative commutative and unital

up to coherent natural isomorphisms on the p ointset level The construction is

joint work of Tony Elmendorf Igor Kriz Mike Mandell and myself and it changes

the nature of stable homotopy theory Ever since its b eginnings with Adams use

of stable homotopy theory to solve the Hopf one problem some thirty

ve years ago most work in the eld has b een carried out working only up to

homotopy formally this means that one is working in the stable homotopy cat

egory For example classically the pro duct on a ring sp ectrum is dened only up

to homotopy and can b e exp ected to b e asso ciative and commutative only up to

homotopy In the new theory we have rings with welldened p ointset level pro d

ucts and they can b e exp ected to b e strictly asso ciative and commutative In the

asso ciative case we call these S algebras The new theory p ermits constructions

that have long b een desired but that have seemed to b e out of reach technically

simple constructions of many of the most basic sp ectra in current use in algebraic

top ology simple constructions of generalized universal co ecient K unneth and

other sp ectral sequences a conceptual and structured approach to Bouseld lo cal

INTRODUCTION

izations of sp ectra a generalized construction of top ological Ho chschild homology

and of sp ectral sequences for its computation a simultaneous generalization of the

algebraic K theory of rings and of spaces etc Working nonequivariantly we shall

describ e the prop erties of the category of S mo dules and shall sketch all but the

last of the cited applications in Chapter XXI I

We return to the equivariant world in Chapter XXI I I which was written jointly

with Elmendorf and Lewis and sketch how the construction of the category of

S mo dules works Here S denotes the sphere Gsp ectrum The starting p oint

G G

of the construction is the twisted half smash pro duct which is a sp ectrum level

generalization of the halfsmash pro duct X n Y X Y of an unbased G

+

space X and a based Gspace Y and is p erhaps the most basic construction in

equivariant stable homotopy theory Taking X to b e a certain Gspace L j of

E of the j linear isometries one obtains a fattened version L j n E

j 1

fold smash pro duct of Gsp ectra Taking j insisting that the E have extra

i

structure given by maps L n E E and quotienting out some of the fat one

i i

obtains a commutative and asso ciative smash pro duct of Gsp ectra with actions

by the monoid L a little adjustment adds in the unit condition and gives the

category of S mo dules The theory had its origins in the notion of an E ring

G 1

sp ectrum intro duced by Quinn Ray and myself over twenty years ago Such rings

j

were dened in terms of op erad actions given by maps L j n E E where

j

E is the j fold smash p ower of E and it turns out that such rings are virtually

the same as our new commutative S algebras The new theory makes the earlier

G

notion much more algebraically tractable while the older theory gives the basic

examples to which the new theory can b e applied

In Chapter XXIV which was written jointly with Greenlees we give a series of

algebraic denitions together with their brave new algebra counterparts and we

show how these notions lead to a general approach to lo calization and completion

theorems in equivariant stable homotopy theory We shall see that Grothendiecks

lo cal cohomology groups are relevant to the study of lo calization theorems in

equivariant homology and that analogs called lo cal homology groups are relevant

to the study of completion theorems in equivariant cohomology We use these

constructions to prove a general lo calization theorem for suitable commutative

S algebras R Taking R to b e the underlying S algebra of R and taking

G G G

M to b e the underlying Rmo dule of an R mo dule M the theorem implies

G G

G

b oth a lo calization theorem for the computation of M BG in terms of M pt

 

INTRODUCTION

 

and a completion theorem for the computation of M BG in terms of M pt

G

Of course this is reminiscent of the AtiyahSegal completion for equivariant K

theory and the Segal conjecture for equivariant cob ordism The general theorem

do es apply to K theory giving a very clean description of K BG but it do es



not apply to cohomotopy there the completion theorem for cohomology is true

but the lo calization theorem for homology is false

We are particularly interested in stable equivariant complex b ordism repre

sented by MU and mo dules over it We explain in Chapter XXI I how simple it

G

is to construct all of the usual examples of MU mo dule sp ectra in the homotopical

sense such as Morava K theory and BrownPeterson sp ectra as brave new p oint

set level MU mo dules We show in Chapter XXI I I how to construct equivariant

versions M as brave new MU mo dules of all such MU mo dules M where G

G G

is any compact Lie group We would like to apply the lo calization theorem of

Chapter XXIV to MU and its mo dule sp ectra but its algebraic hyp otheses are

G

not satised Nevertheless as Greenlees and I explain in chapter XXV the lo

calization theorem is in fact true for MU when G is nite or a nite extension

G

of a The pro of involves the construction of a multiplicative norm map in

G

together with a double coset formula for its computation This dep ends on MU



the fact that MU can b e constructed in a particularly nice way co died in the

G

notion of a global I functor with smash pro duct as a functor of G



These results refo cus attention on stable equivariant complex b ordism whose

study lapsed in the early s In fact some of the most signicant calculational

results obtained then were never fully do cumented in the literature In Chapter

XXVI which was written by Gustavo Comezana new and complete pro ofs of these

results are presented along with results on the relationship b etween geometric

and stable equivariant complex cob ordism In particular when G is a compact

G

Ab elian Lie group Comezana proves that MU is a free MU mo dule on even





degree generators

In Chapter XXVI and in a few places earlier on complete pro ofs are given

either b ecause we feel that the material is inadequately treated in the published

literature or b ecause we have added new material However most of the material

in the b o ok is known and has b een treated in full detail elsewhere Our goal

has b een to present what is known in a form that is more readily accessible and

assimilable with emphasis on the main ideas and the structure of the theory and

with p ointers to where full details and further developments can b e found

INTRODUCTION

Most sections have their own brief bibliographies at the end thus if an authors

work is referred to in a the appropriate reference is given at the end of that

section There is also a general bibliography but since it has over items I felt

that easily found lo cal references would b e more helpful With a few exceptions

the general bibliography is restricted to items actually referred to in the text

and it makes no claim to completeness A full list of relevant and interesting

pap ers would easily double the numb er of entries I oer my ap ologies to authors

not cited who should have b een Inevitably the choice of topics and of material

within topics has had to b e very selective and idiosyncratic

There are some general references that should b e cited here reminders of their

abbreviated names will b e given where they are rst used Starting with Chapter

XI I references to LMS are to

LG Lewis JP May and M Steinb erger with contributions by JE McClure Equivariant

stable homotopy theory Springer Lecture Notes in Mathematics Vol

Most of the material in Chapter XI I and in the ve chapters XVXIX is based

on joint work of Gaunce Lewis and myself that is presented in p erhaps excruciating

detail in that rather encyclop edic volume There are also abbreviated references

in force in particular chapters LL in Chapter XI and tD

in Chapter XVI I

The basic reference for the pro ofs of the claims in Chapters XXI I and XXI I I is

EKMM A Elmendorf I Kriz M A Mandell and J P May Rings mo dules and algebras in

stable homotopy theory Preprint

We shall also refer to the connected sequence of exp ository pap ers

0

A Elmendorf I Kriz M A Mandell and J P May Mo dern foundations for stable EKMM

homotopy theory

GM J P C Greenlees and J P May Completions in algebra and top ology

GM J P C Greenlees and J P May Equivariant stable homotopy theory

These are all in the Handb o ok of Algebraic Top ology edited by Ioan James

that came out in While these have considerable overlap with Chapters XXI I

through XXIV we have varied the p ersp ective and emphasis and each exp osition

includes a go o d deal of material that is not discussed in the other In particular

we p oint to the application of brave new algebra to chromatic p erio dicity in GM

the ideas there have yet to b e fully exploited and are not discussed here

In view of the broad and disparate range of topics we have tried very hard to

make the chapters and often even the sections indep endent of one another We

have also broken the material into short and hop efully manageable chunks only

INTRODUCTION

a few sections are as long as ve pages and all chapters are less than twentyve

pages long Very few readers are likely to wish to read straight through and the

reader should b e unafraid to jump directly to what he or she nds of interest

The reader should also b e unintimidated by nding that he or she has insucient

background to feel comfortable with particular sections or chapters Unfortunately

the sub ject of algebraic top ology is particularly badly served by its textb o oks For

example none of them even mentions lo calizations and completions of spaces

although those have b een standard to ols since the early s We have tried to

include enough background to give the basic ideas Mo dern algebraic top ology is

a thriving sub ject and p erhaps jumping right in and having a lo ok at some of its

more recent directions may give a b etter p ersp ective than trying to start at the

b eginning and work ones way up

As the reader will have gathered this b o ok is a co op erative enterprise Perhaps

this is the right place to try to express just how enormously grateful I am to all

of my friends collab orators and students This b o ok owes everything to our joint

eorts over many years When planning the Alaska conference I invited some of

my friends and collab orators to give talks that would mesh with mine and help

give a reasonably coherent overview of the sub ject Most of the sp eakers wrote up

their talks and gave me license to edit them to t into the framework of the b o ok

Since TeX is refractory ab out listing authors inside a Table of Contents I will here

list those chapters that are written either solely by other authors or jointly with me

Chapter I I I Equivariant rational homotopy theory

by Georgia Triantallou

Chapter VI The homotopy theory of diagrams

by Rob ert Piacenza

Chapter X GCWV complexes and RO Ggraded cohomology

by Stefan Waner

Chapter XI The equivariant Hurewicz and susp ension theorems

by L G Lewis Jr

Chapter XIV An intro duction to equivariant K theory

by J P C Greenlees

Chapter XV An intro duction to equivariant cob ordism

INTRODUCTION

by Steven Costenoble

Chapter XXI Generalized Tate cohomology

by J P C Greenlees and J P May

Chapter XXI I I Brave new equivariant foundations

by A D Elmendorf L G Lewis Jr and J P May

Chapter XXIV Brave new equivariant algebra

by J P C Greenlees and J P May

Chapter XXV Lo calization and completion in complex cob ordism

by J P C Greenlees and J P May

Chapter XXVI Some calculations in complex equivariant b ordism

by Gustavo Costenoble

My deep est thanks to these p eople and to Stefan Jackowski and ChunNip Lee

who also gave talks their topics were the sub jects of their recent excellent survey

pap ers and and were therefore not written up for inclusion here I would

also like to thank Jim McClure whose many insights in this area are reected

throughout the b o ok and Igor Kriz whose collab oration over the last six years has

greatly inuenced the more recent material I would also like to thank my current

Maria Basterra Mike Cole Dan Isaksen Mike Mandell students at Chicago

Adam Przezdziecki Laura Scull and Jerome Wolb ert who have help ed catch

many soft sp ots of exp osition and have already made signicant contributions to

this general area of mathematics

It is an esp ecial pleasure to thank Bob Piacenza and his wife Lyric Ozburn for

organizing the Alaska conference and making it a memorably pleasant o ccasion for

all concerned Thanks to their thoughtful arrangements the intense all day math

ematical activity to ok place in a wonderfully convivial and congenial atmosphere

Finally my thanks to all of those who attended the conference and help ed make

the week such a pleasant mathematical o ccasion thanks for b earing with me

J Peter May

Decemb er

CHAPTER I

ari ant Cellular and Homology Theory Equiv

Some basic denitions and adjunctions

The ob jects of study in equivariant algebraic top ology are spaces equipp ed with

an action by a top ological group G That is the sub ject concerns spaces X to

0 0

x and g g x g g x gether with continuous actions G X X such that ex

Maps f X Y are equivariant if f g x g f x We then say that f is a G

of map The usual constructions on spaces apply equally well in the category GU

Gspaces and Gmaps In particular G acts diagonally on Cartesian pro ducts of

spaces and acts by conjugation on the space MapX Y of maps from X to Y

1

That is we dene g f by g f x g f g x

As usual we take all spaces to b e compactly generated which means that

a subspace is closed if its intersection with each compact Hausdor subspace is

X X is a closed closed and weak Hausdor which means that the diagonal X

where the pro duct is given the compactly generated top ology Among

other things this ensures that we have a G

Map X Y Z MapX Map Y Z

for any Gspaces X Y and Z

H

For us subgroups of G are assumed to b e closed For H G we write X

xjhx x for h H g For x X G fhjhx xg is called the isotropy group f

x

H

of x Thus x X if H is contained in G A go o d deal of the formal homotopy

x

theory of Gspaces reduces to the ordinary homotopy theory of xed p oint spaces

We let NH b e the normalizer of H in G and let W H N H H We sometimes

write N H and W H These Weyl groups app ear ubiquitously in the theory

G G

H

Note that X is a WH space In equivariant theory orbits GH play the role of

I EQUIVARIANT CELLULAR AND HOMOLOGY THEORY

p oints and the set of Gmaps GH GH can b e identied with the group

WH We also have the orbit spaces XH obtained by identifying p oints of X

in the same orbit and these to o are WH spaces For a space K regarded as a

Gspace with trivial Gaction we have

G

U K X GU K X

and

U XG K GU X K

If Y is an H space there is an induced Gspace G Y It is obtained from

H

with g hy for g G h H and y Y A bit G Y by identifying g h y

less obviously we also have the coinduced Gspace Map G Y which is the

H

space of H maps G Y with left action by G induced by the right action of

0 0

G on itself g f g f g g For Gspaces X and H spaces Y we have the

adjunctions

H U Y X GU G Y X

H

and

H U X Y GU X Map G Y

H

Moreover for Gspaces X we have G

GH X G X

H

and

MapGH X Map G X

H

For the rst the unique Gmap G X GH X that sends x X to

H

1

eH x has inverse that sends g H x to the of g g x

A homotopy b etween Gmaps X Y is a homotopy h X I Y that is

a Gmap where G acts trivially on I There results a homotopy category hGU

Recall that a map of spaces is a weak equivalence if it induces an isomorphism

of all homotopy groups A Gmap f X Y is said to b e a weak equivalence

H H H

if f X Y is a weak equivalence for all H G We let hGU denote

the category constructed from hGU by adjoining formal inverses to the weak

equivalences We shall b e more precise shortly The algebraic invariants of G

spaces that we shall b e interested in will b e dened on the category hGU

ANALOGS FOR BASED GSPACES

General References

G E Bredon Intro duction to compact transformation groups Academic Press

Transformation groups Walter de Gruyter T tom Dieck

This reference contains an extensive Bibliography

Analogs for based Gspaces

It will often b e more convenient to work with based Gspaces Basep oints are

We write X for the union of a Gspace Gxed and are generically denoted by

+

X and a disjoint basep oint The wedge or p oint union of based Gspaces is

denoted by X Y The smash pro duct is dened by X Y X Y X Y We

write F X Y for the based Gspace of based maps X Y Then

F X F Y Z F X Y Z

for the category of based Gspaces and we have We write GT

G

T K X GT K X

and

T XG K GT X K

for a based space K and a based Gspace X Similarly for a based Gspace X

and a based H space Y we have

H T Y X GT G Y X

+ H

and

GT X F G Y H T X Y

H +

where F G Y Map G X with the trivial map as basep oint and we have

H + H

Ghomeomorphisms

G X GH X

+ H +

and

F G X F GH X

H + +

A based homotopy b etween based Gmaps X Y is given by a based G

map X I Y Here the based cylinder X I is obtained from X I by

+ +

collapsing the line through the basep oint of X to the basep oint There results a

homotopy category hGT and we construct hGT by formally inverting the weak

I EQUIVARIANT CELLULAR AND HOMOLOGY THEORY

equivalences Of course we have analogous categories hGU and hGU in the

unbased context

In b oth the based and unbased context cobrations and brations are dened

exactly as in the nonequivariant context except that all maps in sight are Gmaps

Their theory go es through unchanged A based Gspace X is nondegenerately

based if the inclusion fg X is a cobration

GCW complexes

n 0

A GCW complex X is the union of sub Gspaces X such that X is a dis

n+1 n

joint union of orbits GH and X is obtained from X by attaching Gcells

n+1 n n

along attaching Gmaps GH S X Such an attaching GH D

n n H

map is determined by its restriction S X and this allows the inductive

analysis of GCW complexes by reduction to nonequivariant homotopy theory

Sub complexes and relative GCW complexes are dened in the obvious way I will

review my preferred way of developing the theory of GCW complexes since this

will serve as a mo del for other versions of cellular theory that we shall encounter

We b egin with the Homotopy Extension and Lifting Prop erty Recall that a map

f X Y is an n equivalence if f is a for q n and a surjection

q

for q n for any choice of basep oint Let b e a function from conjugacy classes

We say that a map e Y Z is of subgroups of G to the

H H H

a equivalence if e Y Z is a H equivalence for all H We allow

H to allow for empty xed p oint spaces We say that a GCW complex

H X has dimension if its cells of orbit typ e GH all have dimension

HELP Let A b e a sub complex of a GCW complex X of Theorem

dimension and let e Y Z b e a equivalence Supp ose given maps g

A Y h A I Z and f X Z such that eg hi and f i hi in

1 0

the following diagram

i i

0 1

o o

A I

A A

x

x

g

h

x

x

x

x

x

x

x

e

o

i i

Z Y

c cF

A

F

~

f A g~

h

F

A

F

A F

o

X I

X X

i i

0 1

GCW COMPLEXES

Then there exist maps g and h that make the diagram commute

n

We construct g and h on A X by induction on n When we pass Proof

from the nskeleton to the n skeleton we may work one cell at a time dealing

with the cells of X not in A By considering attaching maps we quickly reduce

n+1 n

the pro of to the case when X A GH D GH S But this case

n+1 n

reduces directly to the nonequivariant case of D S

Theorem Whitehead Let e Y Z b e a equivalence and X b e a

GCW complex Then e hGU X Y hGU X Z is a bijection if X has



dimension less than and a surjection if X has dimension

Proof Apply HELP to the pair X for the surjectivity Apply HELP to

I for the injectivity the pair X I X

Corollary If e Y Z is a equivalence b etween GCW complexes

of dimension less than then e is a Ghomotopy equivalence

Proof A map f Z Y such that e f id is a homotopy inverse to e



The cellular approximation theorem works equally simply A map f X Y

n n

b etween GCW complexes is said to b e cellular if f X Y for all n and

similarly in the relative case

Approximation Let X A and Y B b e rela Theorem Cellular

0 0

tive GCW complexes X A b e a sub complex of X A and f X A

0 0

Y B b e a Gmap whose restriction to X A is cellular Then f is homotopic

0

rel X A to a cellular map g X A Y B

Proof This again reduces to the case of a single nonequivariant cell

Corollary Let X and Y b e GCW complexes Then any Gmap f

X Y is homotopic to a cellular map and any two homotopic cellular maps

are cellularly homotopic

Proof Apply the theorem in the cases X and X I X I

Theorem For any Gspace X there is a GCW complex X and a weak

X X equivalence

I EQUIVARIANT CELLULAR AND HOMOLOGY THEORY

Proof We construct an expanding sequence of GCW complexes fY ji g

i

together with maps Y X such that jY Cho ose a representative

i i i+1 i i

q H H

map f S X for each element of X x Here q runs over the non

q

negative integers H runs over the conjugacy classes of subgroups of G and x runs

H q

over the comp onents of X Let Y b e the disjoint union of spaces GH S

0

one for each chosen map f and let b e the Gmap induced by the maps f In

0

ductively assume that Y X has b een constructed Cho ose representative

i i

H

maps f g for each pair of elements of Y y that are equalized by

q i q i

here again q runs over the nonnegative integers H runs over the conjugacy classes

H

of subgroups of G and y runs over the comp onents of Y We may arrange that

i

f and g have image in the q skeleton of Y Let Y b e the homotopy co equalizer

i i+1

of the disjoint union of these pairs of maps that is Y is obtained by attaching a

i+1

q

tub e GH S I via each chosen pair f g Dene by use of

+ + i+1

h f g based at y It is easy to triangulate Y as a GCW complex

i i i i+1

that contains Y as a sub complex Taking X to b e the union of the Y and to

i i

b e the map induced by the we obtain the desired weak equivalence

i

The implies that the GCW approximation X is unique

0

up to Ghomotopy equivalence If f X X is a Gmap there is a unique

0 0

f f That homotopy class of Gmaps f X X such that

is b ecomes a functor hGU hGU such that is natural A construction

of that is functorial even b efore passage to homotopy is p ossible Seymour It

follows that the of the category hGU can b e sp ecied by

0 0 0

hGU X X hGU X X hGC X X

where GC is the category of GCW complexes and cellular maps From now on

0

we shall write X X for this set or for its based variant dep ending on the

G

context

Almost all of this works just as well in the based context giving a theory of

GCW based complexes which are required to have based attaching maps This

notion is to b e distinquished from that of a based GCW complex which is just

a GCW complex with a Gxed base vertex In detail a GCW based complex

n 0 n+1

X is the union of based sub Gspaces X such that X is a p oint and X is ob

n n+1

tained from X by attaching Gcells GH D along based attaching Gmaps

+

n n

GH S X Observe that such GCW based complexes are Gconnected

+

in the sense that all of their xed p oint spaces are nonempty and connected

ORDINARY HOMOLOGY AND COHOMOLOGY THEORIES

Nonequivariantly one often starts pro ofs with the simple remark that it suces

to consider connected spaces Equivariantly this wont do many imp ortant foun

dational parts of homotopy theory have only b een worked out for Gconnected

Gspaces

I should emphasize that G has b een an arbitrary top ological group in this dis

cussion When G is a compact Lie group and we shall later restrict attention to

such groups there are imp ortant results saying that reasonable spaces are trian

gulable as GCW complexes or have the homotopy typ es of GCW complexes It

is fundamental for our later work that smo oth compact G are triangula

ble as nite GCW complexes Verona Illman In contrast to the nonequivariant

situation this is false for top ological Gmanifolds which have the homotopy typ es

of GCW complexes but not necessarily nite ones Metric GANRs have the

homotopy typ es of GCW complexes Kwasik Milnors results on spaces of the

homotopy typ e of CW complexes generalize to Gspaces Waner In particular

MapX Y has the homotopy typ e of a GCW complex if X is a compact Gspace

and Y has the homotopy typ e of a GCW complex and similarly for based function

spaces

S Illman The equivariant triangulation theorem for actions of compact Lie groups Math Ann

S Kwasik On the equivariant homotopy typ e of GANRs Pro c Amer Math So c

T Matumoto On GCW complexes and a theorem of JHC Whitehead J Fac Sci Univ of

Tokyo

R M Seymour Some functorial constructions on Gspaces Bull London Math So c

bre bundles Manuscripta Math A Verona Triangulation of stratied

S Waner Equivariant homotopy theory and Milnors theorem Trans Amer Math So c

Ordinary homology and cohomology theories

Let G denote the category of orbit Gspaces GH the standard notation is O

G

Observe that there is a Gmap f GH GK if and only if H is sub conjugate

1

to K since if f eH g K then g H g K Let hG b e the homotopy category of

G Both G and hG play imp ortant roles and it is essential to keep the distinction

in mind

Dene a co ecient system to b e a contravariant functor hG A b One

X of homotopy groups of a based example to keep in mind is the system n

I EQUIVARIANT CELLULAR AND HOMOLOGY THEORY

H

Gspace X X GH X Formally we have an evident xed p oint

n

n

H  K

X induced by a Gmap f GH functor X G T The map X

GK such that f eH g K sends x to g x Any covariant functor hT A b

such as can b e comp osed with this functor to give a co ecient system It

n

should b e intuitively clear that obstruction theory must b e develop ed in terms of

ordinary cohomology theories with co ecients in such co ecient systems The

appropriate theories were intro duced by Bredon

Since the category of co ecient systems is Ab elian with kernels and cokernels

dened termwise we can do in it Let X b e a GCW complex

We have a co ecient system

n n1

C X H X X Z

n n

n H n1 H

That is the value on GH is H X X The connecting homomor

n

H n H n1 H n2

sp ecify a map phisms of the triples X X X

d C X C X

n n1

2

of co ecient systems and d That is we have a of co ecient

systems C X For based GCW complexes we dene C X similarly Write

 

0 0

Hom M M for the Ab elian group of maps of co ecient systems M M

G

and dene

n

X M with Hom d id X M Hom C C

G G

n

G



X M is a co chain complex of Ab elian groups Its homology is the Then C

G



X M Bredon cohomology of X denoted H

G

To dene Bredon homology we must use covariant functors N hG A b

as co ecient systems If M hG A b is contravariant we dene an Ab elian

group

X

N M GH N GH M

G



where the is sp ecied by mf n m f n for a map



f GH GK and elements m M GK and n N GH Here we

write contravariant actions from the right to emphasize the analogy with tensor

pro ducts Such co ends or categorical tensor pro ducts of functors o ccur very

often in equivariant theory and will b e formalized later We dene cellular chains

by

G

X N with d C X N C

G

n n

ORDINARY HOMOLOGY AND COHOMOLOGY THEORIES

G

Then C X N is a chain complex of Ab elian groups Its homology is the Bredon



G

homology of X denoted H X N



Clearly Bredon homology and cohomology are functors on the category GC of

GCW complexes and cellular maps A cellular homotopy is easily seen to induce

a chain homotopy of cellular chain complexes in our Ab elian category of co ecient

systems so homotopic maps induce the same homomorphism on homology and

cohomology with any co ecients

The development of the prop erties of these theories is little dierent from the

nonequivariant case A key p oint is that C X is a pro jective ob ject in the



X is a direct sum of category of co ecient systems To see this observe that C



co ecient systems of the form

n

H H H GK S GK GK

+ +

n 0 0

If F denotes the free Ab elian group functor on sets then

H H

H GK GH H GK F GK F GH GK

0 0 G

0

M GK via M GK In GK M Therefore Hom H

G

GK

0



detail for a Gmap f GH GK we have f f

GK



f F GK GK F GH GK

G G



so that determines via f f This calculation implies the

GK GK

claimed pro jectivity It also implies the dimension

0 

M GK GK M GK M H H

G G

and

G G

N GK H GK N H GK N

 0

these giving isomorphisms of co ecient systems of the appropriate variance as

K varies

X A If A is a sub complex of X we obtain the relative chain complex C



C XA The pro jectivity just proven implies the exp ected long exact sequences



of pairs For additivity just note that the disjoint union of GCW complexes is a

GCW complex For excision if X is the union of sub complexes A and B then

XA as GCW complexes We take the weak equivalence axiom as B B A

a denition That is for general Gspaces X we dene

  G G

H X M H X M and H X N H X N

G G