I EQUIVARIANT CELLULAR AND HOMOLOGY THEORY

Our results on GCW approximation of Gspaces and on cellular approximation

that these are welldened functors on the category hGU Sim of Gmaps imply

ilarly we can approximate any pair X A by a GCW pair X A Less obvi

ously if X A B is an excisive triad so that X is the union of the interiors of A

and B we can approximate X A B by a triad X A B where X is the

union of its sub complexes A and B

That is all there is to the construction of ordinary equivariant homology and

b erg cohomology groups satisfying the evident equivariant versions of the Eilen

Steenro d axioms

G Bredon Equivariant cohomology theories Springer Lecture Notes in Mathematics Vol

S Illman Equivariant singular homology and cohomology Memoirs Amer Math So c No

S J Willson Equivariant homology theories on Gcomplexes Trans Amer Math So c

Obstruction theory

Obstruction theory works exactly as it do es nonequivariantly and Ill just give

n Recall that a connected space X is said to b e nsimple a quick sketch Fix

if X is Ab elian and acts trivially on X for q n Let X A b e a relative

q

H

b e a Gspace such that Y is nonempty connected GCW complex and let Y

n

and nsimple if H o ccurs as an isotropy subgroup of X n A Let f X A Y

n

We ask when f can b e extended to X Comp osing the attaching b e a Gmap

n H

S maps GH X of cells of X n A with f gives elements of Y These

n

elements sp ecify a welldened co cycle

n

C X A Y c

f

n

G

n n

if and only if c A Y and f extends to X If f and f are maps X

f

n

and h is a homotopy rel A of the restrictions of f and f to X A then f f

and h together dene a map

n

Y hf f X I

0

Applying c to cells j I we obtain a deformation co chain

f f h

n

0

Y d C X A

f f h

n

G

0 0

such that Moreo d c c ver given f and d there exists f that coin

f f h f f

n

0

d where the constant homotopy h is cides with f on X and satises d f f

UNIVERSAL COEFFICIENT SPECTRAL SEQUENCES

understo o d This gives the rst part of the following result and the second part

is similar

n n

For f X A Y the restriction of f to X A extends Theorem

n

n

to a map X A Y if and only if c in H X A Y Given maps

f

n

G

n n

f f X Y and a homotopy rel A of their restrictions to X A there is

n

Y that vanishes if and only if the restriction of an obstruction in H X A

n

G

n

the given homotopy to X A extends to a homotopy f f rel A

G Bredon Equivariant cohomology theories Springer Lecture Notes in Mathematics Vol

Universal co ecient sp ectral sequences

While easy to dene Bredon cohomology is hard to compute However we do

t sp ectral sequences which we describ e next have universal co ecien

Let W H b e the comp onent of the identity element of WH and dene a co e

cient system J X by

H

J X GH H X W H Z

X coincides with the obvious co ecient system H X if G is discrete Thus J

We claim that if G is a compact Lie group then J X is the co ecient system

X The p oint is that a Lie theoretic that is obtained by taking the homology of C

argument shows that

H H

GK GK W H

H

We deduce that the ltration of X W H induced by the ltration of X gives rise

X GH to the chain complex C

We can construct an injective resolution Q of the co ecient system M and

C X Q tial the sum of the This is a bicomplex with total dieren form Hom

G

X and of Q It admits two ltrations Using dierentials induced by those of C

pq

comes from the dierential on Q and E one of them the dierential on E

pq

X M Since C X is pro jective the higher Ext groups are zero C is Ext

G

and E reduces to C X M Thus E E H X M and our bicomplex

G G

computes Bredon cohomology Filtering the other way the dierential on E

comes from the dierential on C X and we can identify E Using a pro jective

we obtain an analogous homology sp ectral sequence resolution of N

I EQUIVARIANT CELLULAR AND HOMOLOGY THEORY

Theorem Let G b e either discrete or a compact Lie group and let X b e a

GCW complex There are universal co ecient sp ectral sequences

pq pq

n

H X M X M Ext J E

G

G

and

G G

E Tor J X N H X N

pq pq n

We should say something ab out change of groups and ab out pro ducts in coho

mology but it would take us to o far aeld to go into detail For the rst we simply

note that for H G we can obtain H co ecient systems from Gco ecient sys

G that sends H K to GK G H K For the tems via the functor H

H

second we note that for groups H and G pro jections give a functor from the

orbit category of H G to the pro duct of the orbit categories of H and of G so

that we can tensor an H co ecient system and a Gco ecient system to obtain

an H Gco ecient system When H G we can then apply change of groups

to the diagonal inclusion G G G The resulting pairings of co ecient systems

allow us to dene cup pro ducts exactly as in ordinary cohomology using cellular

approximations of the diagonal maps of GCW complexes

G Bredon Equivariant cohomology theories Springer Lecture Notes in Mathematics Vol

S J Willson Equivariant homology theories on Gcomplexes Trans Amer Math So c

CHAPTER II

ostnikov Systems Lo calization and Completion P

Eilenb ergMacLane Gspaces and Postnikov systems

Let M b e a co ecient system An Eilenb ergMac Lane Gspace K M n is a

Gspace of the homotopy typ e of a GCW complex such that

M if q n

K M n

q

if q n

While our interest is in Ab elian groupvalued co ecient systems we can allow M

to b e setvalued if n and groupvalued if n I will give an explicit construc

tion later Ordinary cohomology theories are characterized by the usual axioms

and by checking the axioms it is easily veried that the reduced cohomology of

based Gspaces X is represented in the form

n

X K M n X M H

G

G

are understo o d where homotopy classes of based maps in hGT

Recall that a connected space X is said to b e simple if A is Ab elian and acts

trivially on X for n More generally a connected space X is said to

n

b e nilp otent if X is nilp otent and acts nilp otently on X for n A

n

H

Gconnected Gspace X is said to b e simple if each X is simple A Gconnected

H

Gspace X is said to b e nilp otent if each X is nilp otent and for each n

H H

the orders of nilp otency of the X groups X have a common b ound

n

We shall restrict our sketch pro ofs to simple Gspaces for simplicity in the next

few sections but everything that we shall say ab out their Postnikov towers and

ab out lo calization and completion generalizes readily to the case of nilp otent G

spaces The only dierence is that each homotopy group system must b e built up

I I POSTNIKOV SYSTEMS LOCALIZATION AND COMPLETION

in nitely many stages rather than all at once

APostnikov system for a based simple Gspace X consists of based Gmaps

X X and p X X

n n n n n

X X for n such that X is a p oint induces an isomorphism

n n

q q

for q n p and p is the Gbration induced from the path space

n n n n

n

bration over a K X K by a map k X n X n It

n

n n

n Our requirement X and that X for q follows that X K

n

q

that Eilenb ergMac Lane Gspaces have the homotopy typ es of GCW complexes

ensures that each X has the homotopy typ e of a GCW complex The maps

n

induce a weak equivalence X lim X but the inverse limit generally will

n n

not have the homotopy typ e of a GCW complex Just as nonequivariantly the

k invariants that sp ecify the tower are to b e regarded as cohomology classes

n n

k H X X

n

n

G

These classes together with the homotopy group systems X sp ecify the weak

n

homotopy typ e of X On passage to H xed p oints a Postnikov system for X

H

gives a Postnikov system for X We outline the pro of of the following standard

result since there is no complete published pro of and my favorite nonequivariant

pro of has also not b een published The result generalizes to nilp otent Gspaces

Theorem A simple Gspace X of the homotopy typ e of a GCW complex

has a Postnikov tower

Pr oof Assume inductively that X X has b een constructed By the

n n

homotopy excision theorem applied to xed p oint spaces we see that the cob er

C is n connected and satises

n

C X

n

n n

More precisely the canonical map F C induces an isomorphism on

n n

for q n We construct

q

j C K X n

n

n

by inductively attaching cells to C to kill its higher homotopy groups We

n

n

take the comp osite of j and the inclusion X C to b e the k invariant k

n n

n

By our denition of a Postnikov tower X must b e the homotopy b er of k

n

x consisting of a path I K Its p oints are pairs X n and a

n

SUMMARY LOCALIZATIONS OF SPACES

n

p oint x X such that and k x The map p X

n n n

X is given by p x x and the map X X is given by

n n n n

j x t x t b eing a p oint on x x x where xt

n n

the cone CX C Clearly p It is evident that induces an

n n n n n

isomorphism on for q n and a diagram chase shows that this also holds for

q

q n

Summary lo calizations of spaces

ariantly lo calization at a prime p or at a set of primes T is a standard Nonequiv

rst step in homotopy theory We quickly review some of the basic theory Say

that a map f X Y is a T cohomology isomorphism if

f H Y A H X A

is an isomorphism for all T lo cal Ab elian groups A

are equivalent Theorem The following prop erties of a nilp otent space Z

When they hold Z is said to b e T lo cal

a Each Z is T lo cal

n

b If f X Y is a T cohomology isomorphism then f Y Z X Z

is a bijection

c The integral homology of Z is T lo cal

Theorem Let X b e a nilp otent space The following prop erties of a map

X X from X to a T lo cal space X are equivalent There is one and up

T T

to homotopy only one such map It is called the lo calization of X at T

a X Z X Z is a bijection for all T lo cal spaces Z

T

b is a T cohomology isomorphism

c X X is lo calization at T

T

d H X Z H X Z is lo calization at T

T

A K Bouseld and D M Kan Homotopy limits completions and lo calizations Springer

Lecture Notes in Mathematics Vol

D Sullivan The genetics of homotopy theory and the Adams conjecture Annals of Math

I I POSTNIKOV SYSTEMS LOCALIZATION AND COMPLETION

Lo calizations of Gspaces

Now let G b e a compact Lie group Say that a Gmap f X Y is a

T cohomology isomorphism if

f H Y M H X M

G G

is an isomorphism for all T lo cal co ecient systems M

The following prop erties of a nilp otent Gspace Z are equivalent Theorem

When they hold Z is said to b e T lo cal

H

a Each Z is T lo cal

b If f X Y is a T cohomology isomorphism then

f Y Z X Z

G G

is a bijection

Theorem Let X b e a nilp otent Gspace The following prop erties of a

map X X from X to a T lo cal Gspace X are equivalent There is one

T T

and up to homotopy only one such map It is called the lo calization of X at T

a X Z X Z is a bijection for all T lo cal Gspaces Z

T

b is a T cohomology isomorphism

H H H

c Each X X is lo calization at T

T

We restrict attention to simple Gspaces Assuming a in Theo Proofs

rem we may replace Z by a weakly equivalent Postnikov tower and we may

assume that the Gspaces X and Y given in b are GCW complexes so that

we are dealing with actual homotopy classes of maps Then a implies b by a

wordforword dualization of our pro of of the Whitehead theorem Conversely b

implies a since the sp ecialization of b to T cohomology isomorphisms of the

f where f X Y is a nonequivariant T cohomology isomor form GH

phism implies b of Theorem In Theorem a implies b by letting Z

run through K M ns and b implies a by Theorem Let Z b e the lo cal

T

ization of Zat T One sees that c implies b by applying the universal co ecient

sp ectral sequence of I taken with homology and co ecient systems tensored

H

with Z The maps induce isomorphisms on homology with co ecients in Z

T T

and one can deduce with some work in the general compact Lie case that they

therefore induce an isomorphism J X Z J X Z Since the universal

T T T

prop erty a implies uniqueness to complete the pro of we need only construct a

SUMMARY COMPLETIONS OF SPACES

map that satises c For this we may assume that X is a Postnikov tower

and we lo calize its terms inductively by lo calizing k invariants and comparing

bration sequences The starting p oint is just the observation that the algebraic

lo calization M M M Z of co ecient systems induces lo calization maps

T T

K M n K M n The relevant diagram is

T

X n X n X K K

X

n

n

n n

K X n K X n

X X

T T

n T n T

n n

We construct the right square by lo calizing the k invariant we dene X to

n T

b e the b er of the lo calized k invariant and we obtain X X making

n n T

the middle square commute and the left square homotopy commute by standard

b er sequence arguments

J P May The dual Whitehead theorems London Math So c Lecture Note Series Vol

J P May J McClure and G V Triantallou Equivariant lo calization Bull London Math

So c

Summary completions of spaces

Completion at a prime p or at a set of primes T is another standard rst step

in homotopy theory Since completion at T is the pro duct of the completions at

p for p T we restrict to the case of a single prime We rst review some of the

nonequivariant theory The algebra we b egin with is a preview of algebra to come

later in our discussion of completions of Gsp ectra at ideals of the Burnside ring

n

A limAp A is neither left nor right exact The padic completion functor

p

and L If in general and it has left derived functors L

F F A

is a free resolution of A then L A and L A are the cokernel and kernel of F

p

F and there results a natural map A L A The higher left derived

p

functors are zero and a short exact sequence

A A A

gives rise to a six term exact sequence

L A L A L A L A L A L A

I I POSTNIKOV SYSTEMS LOCALIZATION AND COMPLETION

If L A then we call A L A the pcompletion of A It must not to b e

confused with the padic completion We say that A is pcomplete if L A and

is an isomorphism The groups L A L A and A are pcomplete for any Ab elian

p

group A While derived functors give the b est conceptual descriptions of L A and

L A there are more easily calculable descriptions Let Zp b e the colimit of the

n n

Zp Z and sequence of homomorphisms p Zp Zp Then Zp

there are natural isomorphisms

HomZp A ExtZp A and L A L A

There is also a natural short exact sequence

n

lim HomZp A L A A

p

A if the ptorsion of A is of b ounded order In particular L A and L A

p

Say that a map f X Y is a pcohomology isomorphism if

f H Y A H X A

is an isomorphism for all pcomplete Ab elian groups A This holds if and only

vector spaces A and this in turn holds if and only if f if it holds for all F

p

H X F H Y F is an isomorphism where F is the eld with p elements

p p p

While this homological characterization is essential to our pro ofs we prefer to

emphasize cohomology

The following prop erties of a nilp otent space Z are equivalent Theorem

When they hold Z is said to b e pcomplete

a Each Z is pcomplete

n

b If f X Y is a pcohomology isomorphism then f Y Z X Z

is a bijection

Theorem Let X b e a nilp otent space The following prop erties of a map

X X from X to a pcomplete space X are equivalent There is one and

p p

up to homotopy only one such map It is called the completion of X at p

a X Z X Z is a bijection for all pcomplete spaces Z

p

b is a pcohomology isomorphism

For n there is a natural and splittable short exact sequence

L X X L X

n n p n

If L X then is also characterized by

COMPLETIONS OF GSPACES

c X X is completion at p

p

A K Bouseld and D M Kan Homotopy limits completions and lo calizations Springer

Lecture Notes in Mathematics Vol

D Sullivan The genetics of homotopy theory and the Adams conjecture Annals of Math

of Gspaces Completions

Now let G b e a compact Lie group Say that a Gmap f X Y is a

pcohomology isomorphism if

f H Y M H X M

G G

is an isomorphism for all pcomplete co ecient systems M This will hold if each

H H H

f X Y is a pcohomology isomorphism by another application of the

universal co ecients sp ectral sequence with a little work in the general compact

Lie case to handle J f

Theorem The following prop erties of a nilp otent Gspace Z are equivalent

When they hold Z is said to b e pcomplete

H

a Each Z is pcomplete

b If f X Y is a pcohomology isomorphism then f Y Z

G

X Z is a bijection

G

Theorem Let X b e a nilp otent Gspace The following prop erties of a

map X X from X to a pcomplete Gspace X are equivalent There is

p p

one and up to homotopy only one such map It is called the completion of X

at p

a X Z X Z is a bijection for all pcomplete Gspaces Z

p

b is a pcohomology isomorphism

H H H

c Each X X is completion at p

p

For n there is a natural short exact sequence

L X X L X

p

n n n

Proofs The pro ofs are the same as those of Theorems and except that

completions of Eilenb ergMac Lane Gspaces are not Eilenb ergMac Lane Gspaces

in general For a co ecient system M M L M induces pcompletions

K M n K L M n whenever L M For the general case let FM b e

I I POSTNIKOV SYSTEMS LOCALIZATION AND COMPLETION

the co ecient system obtained by applying the free Ab elian group functor to M

regarded as a setvalued functor There results a natural epimorphism FM M

of co ecient systems Let F M b e its kernel Since L vanishes on free mo dules

we can construct the completion of K M n at p via the following diagram of

brations

K M n K F M n K F M n K F M n

K M n

K L F M n K L F M n K L F M n

p

That is K M n is the b er of K L F M n K L F M n It is

p

complete since its homotopy group systems are complete The map K M n

K M n is a pcohomology isomorphism b ecause its xed p oint maps are so by

p

the Serre sp ectral sequence

J P May Equivariant completion Bull London Math So c

CHAPTER III

ariant Rational Homotopy Theory Equiv

by Georgia Triantallou

Summary the theory of minimal mo dels

Let G b e a nite group In this chapter we summarize our work on the alge

braicization of rational Ghomotopy theory

To simplify the statements we assume simply connected spaces throughout the

chapter The theory can b e extended to the nilp otent case in a straightforward

manner We recall that by rationalizing a space X we approximate it by a space

X the homotopy groups of which are equal to X Q thus neglecting the tor

sion The advantage of doing so is that rational homotopy theory is determined

and later by Sullivan completely by algebraic invariants as was shown by Quillen

Our theory is analogous to Sullivans theory of minimal mo dels which we now re

view For our purp oses we prefer Sullivans approach b ecause of its computational

advantage and its relation to geometry by use of dierential forms

The algebraic invariants that determine the rational homotopy typ e are certain

algebras that we call DGAs By denition a DGA is a graded commutative

n n

asso ciative algebra with unit over the rationals with dierential d A A

for n We say that A is c onnected if H A Q and simply connected if in

addition H A Again we assume that all DGAs in sight are connected and

simply connected A map of DGAs is said to b e a quasiisomorphism if it induces

an isomorphism on cohomology

Certain DGAs the so called minimal ones play a sp ecial role to b e describ ed

is said to b e minimal if it is free and its dierential is decom b elow A DGA M

I I I EQUIVARIANT RATIONAL HOMOTOPY THEORY

p osable Freeness means that M is the tensor pro duct of a p olynomial algebra

generated by elements of even degree and an exterior algebra generated by elements

M M of o dd degree Decomp osability of the dierential means that dM

where M is the set of p ositive degree elements of M

There is an algebraic notion of homotopy b etween maps of DGAs that mirrors

the top ological notion Let Q t dt b e the free DGA on two generators t and dt

of degree and resp ectively with dt dt

Definition Two morphisms f g A B are homotopic if there is a map

H f and e H g where e is the H A B Q t dt such that e

pro jection t dt and e the pro jection t dt

The basic example of a DGA in the theory is the PL De Rham algebra E of a

X

simplicial complex X which is constructed as follows Let

n n n

f t t t j t t g

n i i

i

n

A p olynomial form of degree b e an nsimplex of X canonically emb edded in R

n

p on is an expression

X

f t t dt dt

I n i i

p

1

I

i g and f is a p olynomial with co ecients in Q A global where I fi

p I

PL piecewise linear form on X is a collection of p olynomial forms one for each

simplex of X which coincide on common faces The set of PL forms of X is the

DGA E A version of the classical de Rham theorem holds namely that

X

H E H X Q

X

We have the following facts

Theorem A quasiisomorphism b etween minimal DGAs is an isomor

phism

Theorem If f A B is a quasiisomorphism of DGAs and M is a

minimal DGA then f M A M B is an isomorphism

Theorem For any simply connected DGA A there is a minimal DGA M

and a quasiisomorphism M A Moreover M is unique up to noncanonical

isomorphism namely if M A is another quasi isomorphism then there is

an isomorphism e M M such that e and are homotopic

EQUIVARIANT MINIMAL MODELS

Here M is said to b e the minimal model of A The minimal mo del M of

X

the PL de Rham algebra E of a simply connected space X is called the minimal

X

mo del of X

Theorem The corresp ondance X M induces a bijection b etween ra

X

tional homotopy typ es of simplicial complexes on the one hand and isomorphism

classes of minimal DGAs on the other

More precisely assuming X is a rational space the homotopy groups X of

n

X are isomorphic to QM where QM M M M is the space of

X n

indecomp osables of M The nth stage X of the Postnikov tower of X has M n

n X

as its minimal mo del where Mn denotes the subalgebra of M that is generated

n n

by the elements of degree n The k invariant k H X X which

n n

n

can b e represented as a map X H X is determined by the dif

n n

n

ferential d QM H M n These prop erties enable the inductive

X n X

construction of a rational space that realizes a given minimal algebra

On the morphism level we have

Theorem If Y is a rational space then

X Y M M

Y X

We warn here that the minimal mo del though very useful computationally is

not a functor In particular a map of spaces induces a map of the corresp onding

minimal mo dels only up to homotopy

D Quillen Rational Homotopy Theory Ann of Math

Math IHES D Sullivan Innitesimal Computations in Top ology Publ

Equivariant minimal mo dels

For nite groups G an analogous theory can b e develop ed for Grational homo

topy typ es of Gsimplicial complexes For simplicity we assume throughout that

the spaces X are Gconnected and Gsimply connected which means that each

H

xed p oint space X is connected and simply connected however the theory

works just as well for Gnilp otent spaces In fact by work of B Fine the the

ory can b e extended in such a way that no xed base p oint and no connectivity

assumption on the xed p oint sets are required

Let V ec b e the category of rational co ecient systems and V ec the category of

G

G

to rational vector spaces Our invariants for determining covariant functors from G

I I I EQUIVARIANT RATIONAL HOMOTOPY THEORY

Grational homotopy typ es are functors of a sp ecial typ e from G into DGAs which

we now describ e

Definition A system of DGAs is a covariant functor from G to simply

connected DGAs such that the underlying functor in V ec is injective

G

The injective ob jects of V ec or equivalently the pro jective rational co ecient

G

systems can b e characterized as follows

Theorem i For H G and a WH representation V there is a pro jective

co ecient system V V ec such that

G

K

GK Q GH V V

Q W H

K

where the rst factor is the vector space generated by the set GH

ii Every pro jective co ecient system is a direct sum of such V s

H

The basic system of DGAs in the theory is the system of de Rham algebras E

X

H

of the xed p oint sets X of a Gsimplicial complex X We denote this system

It is crucial to realize that E is injective and that this prop erty is central by E

X X

can b e shown by utilising the splitting of X to the theory The injectivity of E

X

into its orbit typ es

We note that E together with the induced Gaction determine a minimal al

X

gebra equipp ed with a Gaction However there are in general many Grational

homotopy typ es of Gsimplicial complexes that realize this minimal Galgebra In

order to have unique spacial realizations we need to take into account the algebraic

data of all xed p oint sets which leads us to systems of DGAs

Dene the cohomology of a system A of DGAs with resp ect to a covariant co e

to b e the cohomology of the co chain complex H om cient system N V ec N A

G

G

An equivariant de Rham theorem follows by use of the universal co ecients sp ec

tral sequence

Theorem For M V ec with dual M V ec

G

G

M E X M H H

X

G G

The lack of functoriality of the minimal mo del of a space complicates the con

struction of equivariant minimal mo dels We cannot for instance dene the

system of minimal mo dels M H of the xed p oint sets of a Gcomplex X It

X

turns out that the right denition of minimal mo dels in the equivariant context is

the following

EQUIVARIANT MINIMAL MODELS

Definition A system M of DGAs is said to b e minimal if

i each algebra MGH is free commutative

ii the DGA MGG is minimal and

iii the dierential on each MGH is decomp osable when restricted to the

intersection of the kernels of the maps MGH MGK induced by

prop er inclusions H K

One can think of ii as an initial condition and of iii as the minimality

condition that guarantees the uniqueness of equivariant minimal mo dels As in

the nonequivariant case minimal systems are classied by their cohomology

Theorem A quasiisomorphism b etween minimal systems of DGAs is an

isomorphism

Also Theorems and have equivariant counterparts

Theorem If A is a system of DGAs then there is a quasiisomorphism

f M A where M is a minimal system Moreover M is unique up to non

canonical isomorphism

This result provides the existence of equivariant minimal mo dels Unlike the

nonequivariant case the pro of is rather involved and is based on a careful inves

tigation of the universal co ecients sp ectral sequence We dene the equivariant

minimal model M of a Gsimplicial complex X to b e the minimal system of

X

DGAs that is quasiisomorphic to the system of de Rham algebras E

X

A notion of homotopy can b e dened for systems of DGAs If A is a system of

DGAs we denote by A Qt dt the functor

A Q t dtGH AGH Qt dt

It can b e shown that this functor is injective and therefore it forms a system of

DGAs Homotopy of maps of systems of DGAs can now b e dened in the obvious

way suggested by the nonequivariant case Let A B denote homotopy classes

G

of maps of systems

Theorem If f A B is a quasiisomorphism of systems of DGAs and

M is a minimal system of DGAs then

f M A M B

G G

is an isomorphism

I I I EQUIVARIANT RATIONAL HOMOTOPY THEORY

The equivariant minimal mo del determines the rational Ghomotopy typ e of a

Gspace namely

The corresp ondence X M induces a bijection b etween ra Theorem

X

tional Ghomotopy typ es of Gsimplicial complexes on the one hand and isomor

phism classes of minimal systems of DGAs on the other

More precisely there is a ltration of minimal subsystems of DGAs

M n M n M

X X X

such that each stage is the equivariant minimal mo del of the equivariant Postnikov

term X of the space X The system of rational homotopy groups of the xed

n

p oint sets X Q and the rational equivariant k invariants can also b e read

n

from the mo del M This makes the inductive construction of the Postnikov

X

decomp osition of the rationalization X p ossible if the equivariant minimal mo del

is given

On the morphism level we have

If Y is a rational Gsimplicial complex then there is a bijection Theorem

X Y M M

Y X

Aquivariante Rationale Homotopietheorie Bonner Math Schriften Vol G Triantallou

G Triantallou Equivariant minimal mo dels Trans Amer Math So c

equivariant Hopf spaces Rational

In spite of the conceptual analogy of the equivariant theory to the nonequivariant

one the calculations in the equivariant case are much more subtle and can yield

surprising results We illustrate this by describing our work on rational Hopf G

spaces It is a basic feature of nonequivariant homotopy theory that the rational

Hopf spaces split as pro ducts of Eilenb ergMac Lane spaces The equivariant

analogue is false By a Hopf Gspace we mean a based Gspace X together with a

Gmap X X X such that the base p oint is a twosided unit for the pro duct

Examples include Lie groups K with a Gaction such that G is a nite subgroup

of the inner automorphisms of K and lo op spaces X of Gspaces based at a

Gxed p oint of X

RATIONAL EQUIVARIANT HOPF SPACES

Theorem Let X b e a Gconnected rational Hopf Gspace of nite typ e

If G is cyclic of prime p ower order then X splits as a pro duct of Eilenb erg

Mac Lane Gspaces If G Z Z for distinct primes p and q then there are

p q

counterexamples to this statement

Outline of pro of In this outline we suppress the technical part of the pro of

th

which is quite extensive As in the nonequivariant case the n term X of

n

n

a Postnikov tower of X is a Hopf Gspace Moreover the k invariant k

n

X is a primitive element This means that H X

n

n

n n n

m k p k p k

n

X where m is the pro duct and the p are the pro jections in H X X

i n n

n

and Z Z stems from the fact that rational The dierence in the two cases Z k

p q

p

co ecient systems for these groups have dierent pro jective dimensions Indeed

systems for Z k have pro jective dimension at most whereas there are rational

p

co ecient systems for Z Z of pro jective dimension Using this fact ab out

p q

Z k we can compute inductively the equivariant minimal mo del of each Postnikov

p

term X and its cohomology In particular we show that all nonzero elements of

n

n

H X X are decomp osable and therefore nonprimitive

n

n

G

In the case of Z Z we construct counterexamples which are stage Postnikov

p q

systems with primitive k invariant As in the nonequivariant case if X has only

two nonvanishing homotopy group systems then the primitivity of the unique k

invariant is a sucient condition for X to b e a Hopf Gspace By construction the

two systems of homotopy groups X and X are as follows The groups

n n

H Z Z

p q

Z The groups X are zero for all prop er subgroups H and X

n n

H

X are zero for all nontrivial subgroups H and X Z The rst

n n

co ecient system has pro jective dimension This and the universal co ecients

n

X Z Moreover all nonzero elements X sp ectral sequence yields H

n

n

G

of this group are primitive This gives an innite choice of primitive k invariants

and therefore an innite collection of rationally distinct Hopf Gspaces which do

not split rationally into pro ducts of Eilenb ergMac Lane Gspaces

The counterexamples X constructed in the theorem are innite lo op Gspaces

in the sense that there are Gspaces E and homotopy equivalences E E

n n n

with X E For the more sophisticated notion of innite lo op Gspaces where

indexing over the representation ring of G is used no such pathological b ehavior is p ossible

I I I EQUIVARIANT RATIONAL HOMOTOPY THEORY

As a nal comment we mention that the theory of equivariant minimal mo dels

has b een used by my collab orators and myself to obtain aplications of a more

geometric nature like the classication of a large class of Gmanifolds up to nite

ambiguity and the equivariant formality of GK ahler manifolds

M Rothenb erg and G Triantallou On the classication of Gmanifolds up to nite ambiguity

Comm in Pure and Appl Math

B Fine and G Triantallou Equivariant formality of GKahler manifolds Canadian J Math

To app ear

G Triantallou Rationalization of Hopf Gspaces Math Zeit

CHAPTER IV

Smith Theory

cohomology Smith theory via Bredon

We shall explain two approaches to the classical results of PA Smith We b egin

with the statement Let G b e a nite pgroup and let X b e a nite dimensional

GCW complex such that H X F is a nite dimensional vector space where

p

F denotes the eld with p elements All cohomology will have co ecients in F

p p

here

G

Theorem If X is a mo d p cohomology nsphere then X is empty or is a

mo d p cohomology msphere for some m n If p is o dd then n m is even and

G

X is nonempty if n is even

G H GH

If H is a nontrivial normal subgroup of G then X X By induction

on the order of G Theorem will b e true in general if it is true when G Zp is

the cyclic group of order p Our rst pro of is an almost trivial exercise in the use

of Bredon cohomology We restrict attention to G Zp but we do not assume

that X is a mo d p cohomology sphere until we put things together at the end

Observe that an exact sequence

L M N

of co ecient systems give rise to a long exact sequence

q q q q

X L X N H X M H X L H H

G G G G

G

Let FX XX The action of G on FX is free away from the basep oint There

are co ecient systems L M and N such that

IV SMITH THEORY

q

q

H F XG X L H

G

q

q

H X M H X

G

and

q

q G

H X X N H

G

To determine L M and N we need only calculate the right sides when q and

X is an orbit that is X G or X We nd

LG F L

p

M G F G M F

p p

N G N F

p

n

Let I b e the augmentation ideal of the group ring F G and let I denote b oth

p

th n

the n p ower of I and the co ecient system whose value on G is I and whose

p

value on is zero Then I L It is easy to check that we have exact sequences

of co ecient systems

I M L N

and

L M I N

These exact sequences coincide if p By they give rise to long exact

sequences

q q

q q q G

X I X I H X H F XG H X H H

G G

and

q

q q q G q

H X I H H F XG H X H X F XG

G

Dene

q

q q q G

a X I b dim H F XG a dim H dim H X c dim H X

q q q q

G

Note that a a if p We read o the inequalities

q q

a c b a and a c b a

q q q q q q q q

Iteratively these imply the following inequality for q and r

b b b a a c c c

q q q r q r q q q q r

BOREL COHOMOLOGY LOCALIZATION AND SMITH THEORY

where r is o dd if p In particular with q and r large

X X

c b

q q

Using the further short exact sequences

n n

I I L n p

we can also read o the the Euler characteristic formula

G

X X p F XG

P P

First proof of Theorem Here b hence c The case

q q

P

G

c is ruled out by the congruence X X mo d p when p this

q

G

congruence also implies that n m is even and that X is nonempty if n is even

Taking q n and r large in we see that m cannot b e greater than n

J P May A generalization of Smith theory Pro c Amer Math So c

P A Smith Transformations of nite p erio d Annals of Math

lo calization and Smith theory Borel cohomology

Let EG b e a free contractible Gspace For a Gspace X the Borel construction

on X is the orbit space EG X and the Borel homology and cohomology of

G

X with co ecients in an Ab elian group A are dened to b e the nonequivariant

homology and cohomology of this space For reasons to b e made clear later the

Borel construction is also called the homotopy orbit space and is sometimes

denoted X People not fo cused on equivariant algebraic top ology very often

hG

refer to Borel cohomology as equivariant cohomology We can relate it to Bredon

denote the constant co ecient system at A cohomology in a simple way Let A

Since the orbit spaces GH G are p oints we see immediately from the axioms

is isomorphic to H XG A and similarly in homology Therefore X A that H

G

q

EG X A H H EG X A H EG X A and H EG X A

G G

G G

Observe that the Borel cohomology of a p oint is the cohomology of the classifying

space BG E GG In this section we shall use the notation

H X H EG X

G

G

standard in much of the literature

IV SMITH THEORY

Here we x a prime p and understand mo d p co ecients If X is a based Gspace

we let H X b e the kernel of H X H H BG Equivalently

G G G

H X H EG X

G

G

Because G acts freely on EG it acts freely on EG X Therefore by the

Whitehead theorem if f X Y is a Gmap b etween GCW complexes that is

a nonequivariant homotopy equivalence then

f EG X EG Y

is a Ghomotopy equivalence and therefore

f EG X EG Y

G G G

is a homotopy equivalence At rst sight it seems unreasonable to exp ect EG X

G

G

to carry much information ab out X but it do es

n

We now assume that G is an elementary Ab elian pgroup G Zp for some

n and that X is a nite dimensional GCW complex We shall describ e how to

G

use Borel cohomology to determine the mo d p cohomology of X as an algebra

over the Steenro d algebra and we shall sketch another pro of of Theorem Our

starting p oint is the lo calization theorem

n

Since G Zp H BG is a p olynomial algebra on n generators of degree

one if p and is the tensor pro duct of an exterior algebra on n generators of

degree one and the p olynomial algebra on their Bo cksteins if p Let S b e the

multiplicative subset of H BG generated by the nonzero elements of degree one

if p and by the nonzero images of Bo cksteins of degree two if p

Theorem Localiza tion For a nite dimensional GCW complex X

G

the inclusion i X X induces an isomorphism

G

X X S H i S H

G G

G G

Pro of Let FX XX By the cob er sequence X X FX

it suces to show that S H FX Here FX is a nite dimensional G

G

G

CW complex and FX By induction up skeleta it suces to show that

q

S H Y when Y is a wedge of copies of GH S for some H G

G

and such a wedge can b e rewritten as Y GH K where K is a wedge of

q

copies of S Since EG GH E GH is a mo del for BH we see that

G

EG Y BH K At least one element of S restricts to zero in H BH

G

and this implies that S H Y G

BOREL COHOMOLOGY LOCALIZATION AND SMITH THEORY

Lo calization theorems of this general sort app ear ubiquitously in equivariant

theory As here the pro ofs of such results reduce to the study of orbits by general

nonsense arguments and the sp ecics of the situation are then used to determine

what happ ens on orbits When n we can b e a little more precise

q q

G

X is an X H Lemma If G Zp and dim X r then i H

G G

isomorphism for q r

Proof It suces to show that H FX for q r Since FX is G

G

free away from its basep oint the pro jection EG S induces a Ghomotopy

equivalence EG FX FX and therefore a homotopy equivalence EG

G

FX F XG Obviously dimF XG r

G G G

Since G acts trivially on X EG X BG X

G

Second proof of Theorem Take G Zp and let X b e a mo d p ho

G

mology nsphere We assume that X is nonempty The Serre sp ectral sequence

of the bundle EG X BG converges from

G

H G H X H BG H X

X X Since a xed p oint of X gives a section E E Therefore H to H

G G

is a free H BGmo dule on one generator of degree n and in high degrees this

must b e isomorphic to

G G G

X H BG X H BG H X H

G

G

By a trivial dimension count this can only happ en if X is a mo d p cohomology

msphere for some m Naturality arguments from the H BGmo dule structure

show that m must b e less than n and must b e congruent to n mo d if p

G G

To see that X is nonempty if p and n is even one assumes that X is

empty and deduces from the multiplicative structure of the sp ectral sequence that

X cannot b e nite dimensional

Returning to the context of the lo calization theorem one would like to retrieve

G

X As a matter of algebra S H X inherits H X algebraically from S H

G G

a structure of algebra over the mo d p Steenro d algebra A from H X However

G

it no longer satises the instability conditions that are satised in the cohomology

IV SMITH THEORY

of spaces For any Amo dule M the subset of elements that do satisfy these

conditions form a submo dule UnM Obviously the lo calization map

G G G

S H X H X S H X H BG H X

G G G

takes values in UnS H X By a purely algebraic analysis using basic infor

G

mation ab out the Steenro d op erations Dwyer and Wilkerson proved the following

remarkable result They assume that X is nite but the argument still works

when X is nite dimensional

Theorem For any elementary Ab elian pgroup G and any nite dimen

sional GCW complex X

G

H BG H X UnS H X

G

is an isomorphism of Aalgebras and H BGmo dules Therefore

G



H X F UnS H X

p

H B G

G

We will come back to this p oint when we talk ab out the Sullivan conjecture

A Borel et al Seminar on transformation groups Annals of Math Studies Princeton

GE Bredon Intro duction to compact transformation groups Academic Press

Transformation groups Walter de Gruyter T tom Dieck

WG Dwyer and CW Wilkerson Smith theory revisited Annals of Math

WY Hsiang Cohomology theory of top ological transformation groups Springer

CHAPTER V

Constructions Equivari ant Applications Categorical

Co ends and geometric realization

We pause to intro duce some categorical and top ological constructs that are used

ubiquitously in b oth equivariant and nonequivariant homotopy theory They will

b e needed in a numb er of later places We are particularly interested in homotopy

colimits These are examples of geometric realizations of spaces which in turn are

examples of co ends which in turn are examples of co equalizers

b e a small category and let C b e a category that has all colimits Write Let

for the categorical copro duct in C The co equalizer C f f of maps f f X

Y is a map g Y C f f such that g f g f and g is universal with this

prop erty It can b e constructed as the pushout in the following diagram where

r is the folding map

0

f f

X X

Y

g

r

C f f

X

Co ends are categorical generalizations of tensor pro ducts Given a functor F

op

C the co end

Z

F n n

is dened to b e the co equalizer of the maps

a a

F n m F n n f f

mn

V CATEGORICAL CONSTRUCTIONS EQUIVARIANT APPLICATIONS

whose restrictions to the th summand are

F id F n m F m m and F I d F n m F n n

resp ectively It satises a universal prop erty like that of tensor pro ducts If the

ob jects of F n n have p oints that can b e written in the form of tensors x y

then the co end is obtained from the copro duct of the F n n by identifying x y

with x y whenever this makes sense Here is a map in contravariant actions

are written from the right and covariant actions are written from the left

op

Dually if C has limits a functor F C has an end

Z

F n n

It is dened to b e the equalizer E f f of the maps

Y Y

f f F n F m n

mn

whose pro jections to the th factor are

F id F m m F m n and F id F n n F m n

Recall that a simplicial ob ject in a category C is a contravariant functor

C where is the category of sets n f ng and monotonic maps Using

U the usual face and degeneracy maps we obtain a covariant functor M

to the standard top ological nsimplex For a simplicial space that sends n

n

X U we have the pro duct functor

op

X M U

Dene the geometric realization of X to b e the co end

Z

jX j X

n n

If X is a simplicial based space so that all its face and degeneracy maps are

basep oint preserving then all p oints of each subspace fg are identied to

n

the p oint X in the construction of jX j hence

Z

jX j X

n n

If X is a simplicial Gspace then jX j inherits a Gaction such that

H H

jX j jX j for all H G

HOMOTOPY COLIMITS AND LIMITS

S Mac Lane Categories for the Working Mathematician Springer

Springer Lecture Notes Vol J P May The Geometry of Iterated Lo op Spaces x

Homotopy colimits and limits

b e any small top ological category We understand D to have a discrete Let D

ob ject set and to have spaces of maps d d such that comp osition is continuous

Let B D b e the set of ntuples f f f of comp osable arrows of D

n n

depicted

f f f

n

1 2

o o o o

d d d

n

Here B D is the set of ob jects of D and B D is top ologized as a subspace of the

n

nfold pro duct of the total morphism space D d d With zeroth and last face

given by deleting the zeroth or last arrow of ntuples f or by taking the source or

target of f if n and with the remaining face and degeneracy op erations given

by comp osition or by insertion of identity maps in the appropriate p osition B D

is a simplicial set called the nerve of D Its geometric realization is the classifying

space B D If D has a single ob ject d then G D d d is a top ological monoid

asso ciative Hopf space with unit and B D BG is its classifying space

We can now dene the twosided categorical bar construction It will sp ecialize

to give homotopy colimits Let T D U b e a continuous contravariant

functor This means that each T d is a space and each function T D d d

U T d T d is continuous Let S D U b e a continuous covariant functor

We dene

B T D S jB T D S j

Here B T D S is the simplicial space whose set of nsimplices is

ft f sjt T d f B D and s S d g

n n

n

top ologized as a subspace of the pro duct T d D d d S d

B T D S T d S d The zeroth and last face use the evaluation of the

functors T or S the remaining faces and the degeneracies are dened like those of

B D

op

Since the co end of T S D D U is exactly the co equalizer of d d

B T D S B T D S we obtain a natural map

Z

D

B T D S T d S d T S D

V CATEGORICAL CONSTRUCTIONS EQUIVARIANT APPLICATIONS

It is obtained by using iterated comp ositions to mapB T D S to the constant

simplicial space at the cited co end which we denote by T S

D

b e the covariant functor represented by an ob ject e of D so that D d Let D

e e

D e d Then reduces to a map

B T D D T e

e

and this map is a homotopy equivalence In fact using the identity map of e we

obtain an inclusion T e B T D D such that and a simplicial

e

deformation id There is a leftrigh t symmetric analogue

If the functor S takes values in GU then B T D S is a simplicial Gspace

and B T D S is a Gspace such that

H H

B T D S B T D S

We dene the homotopy colimit of our covariant functor S by

Ho colim S B D S

where D U is the trivial functor to a p oint space Here the co end on

the right of is exactly the ordinary colimit of S Thus we have

ho colim S colim S

When G is a group regarded as category with a single ob ject and X is a left G

space regarded as a covariant functor the homotopy colimit of X is the homotopy

orbit space EG X XhG and is the natural map XhG XG

G

Our preferred denition of homotopy limits is precisely dual We have a cosim

plicial space C T D S the twosided cobar construction Its set of ncosimplices

B D of the spaces T d S d top ologized as a is the pro duct over all f

n n

subspace of Map B D T d S d The f th co ordinates of the cofaces and

n

co degeneracies with target C T D S are obtained by pro jecting onto the co

n

ordinate of their source that is indexed by the corresp onding face or degeneracy

except that for the zeroth and last coface we must comp ose with applied to f

T f id T d S d T d S d

n n

or

id S f T d S d T d S d

n n n

HOMOTOPY COLIMITS AND LIMITS

We dene the geometric realization or totalization Tot Y of a cosimplicial

space Y to b e the end

Z

Map Y Tot Y

n n

op

Here we are using the evident functor U that sends m n to

Map Y If Y takes values in based spaces we may rewrite this as

m n

Z

Tot Y F Y

n n

We then dene

C T D S TotC T D S

and we have a natural map

Z

D

T d S d C T D S

We dene the homotopy limit of our contravariant functor T D U to b e

Holim T Tot C T D

and we see that sp ecializes to give a natural map

lim T holim T

When G is a group regarded as a category with a single ob ject and X is a

right Gspace regarded as a contravariant functor the homotopy limit of X is

the homotopy xed p oint space of Gmaps EG X

G hG

Map E G X MapE G X X

G

G hG

and is the natural map X X that sends a xed p oint to the constant

function at that p oint This map is the ob ject of study of the Sullivan conjecture

A K Bouseld and D M Kan Homotopy limits completions and lo calizations Springer

Lecture Notes in Mathematics Vol

J P May Classifying spaces and brations x Memoirs Amer Math So c No

V CATEGORICAL CONSTRUCTIONS EQUIVARIANT APPLICATIONS

Elmendorf s theorem on diagrams of xed p oint spaces

Recall that G is the category of orbit spaces We shall regard G as a top ological

category with a discrete set of ob jects We write GH for a typical ob ject to

avoid confusing it with the Gspace GH The space of morphisms GH

GK is the space of Gmaps GH GK and this space may b e identied

H

with GK Dene a G space to b e a continuous contravariant functor G

U A map of G spaces is a natural transformation and we write GU for the

category of Gspaces We shall compare this category with GU We have already

observed that a Gspace X gives a G space X and we write

GU GU

for the functor that sends X to X We wish to determine how much information

the functor loses

By the denition of C X it is clear that the ordinary homology and coho

mology of X dep end only on X If T G U is a G space such that each

T GH is a CWcomplex and each T GK T GH is a cellular map then

X M Note however that T M exactly as we dened H we can dene H

G G

H

unless G is discrete X will not inherit a structure of a CWcomplex from a

GCW complex X Indeed for compact Lie groups we saw that it was not quite

the functor X that was relevant to ordinary cohomology but rather the functor

H

that sends GH to X W H

There is an obvious way that G spaces determine Gspaces

GU GU by T T Ge with the Lemma Dene a functor

Gmaps Ge Ge inducing the action Then is left adjoint to

GU T X GU T X

Proof Clearly X X The quotient map G GH induces a map

H

T GH T Ge and these maps together sp ecify a natural map

T T Passage from T X to T X is a bijection whose

inverse sends f T X to f

The following result of Elmendorf shows that G spaces determine Gspaces in a

less obvious way In fact up to homotopy any G space can b e realized as the xed

p oint system of a Gspace and up to homotopy the functor has a right adjoint

as well as a left adjoint Note that we can form the pro duct T K of a G space

ELMENDORFS THEOREM ON DIAGRAMS OF FIXED POINT SPACES

T and a space K by setting T K GH T GH K In particular T I

is dened and we have a notion of homotopy b etween maps of G spaces Write

T T for the set of homotopy classes of maps T T

G

There is a functor GU GU and a Theorem Elmendorf

H

natural transformation id such that each T T GH is

a homotopy equivalence If X has the homotopy typ e of a GCW complex then

there is a natural bijection

X T X T

G G

Proof Let S G GU b e the covariant functor that sends the ob ject

GH to the Gspace GH On morphisms it is given by identity maps

G GH GK GU GH GK

For a G space T dene T to b e the Gspace B T G S We have

H H

S GK GK GU GH GK G GH GK

H

and and give homotopy equivalences T T GH that dene

a natural transformation id Clearly

T T T

is a weak equivalence of Gspaces for any T With T X this gives a weak

equivalence X X We can check that X has the homotopy typ e

of a GCW complex if X do es Therefore is an equivalence and we cho ose a

homotopy inverse Dene

X T X T and X T X T

G G G G

by f f and Easy diagram chases show that

and f f Since is a weak equivalence the

Whitehead theorem gives that is a bijection It follows formally that and

are inverse bijections

A D Elmendorf Systems of xed p oint sets Trans Amer Math So c

V CATEGORICAL CONSTRUCTIONS EQUIVARIANT APPLICATIONS

Eilenb ergMac Lane Gspaces and universal F spaces

We give some imp ortant applications of this construction starting with the

construction of equivariant Eilenb ergMac Lane spaces that we promised earlier

Let B b e the classifying space functor from top ological monoids Example

to spaces It is pro ductpreserving and it therefore gives an Ab elian top ological

group when applied to an Ab elian top ological group If is a discrete Ab elian

n

group then the nfold iterate B is a K n A co ecient system M hG

A b may b e regarded as a continuous functor G U with discrete values We

n n

may comp ose with B to obtain a G space B M In view of the equivalences

n H n

B M K M GH n B M is a K M n Theorem gives

a homotopical description of ordinary cohomology in terms of maps of G spaces

n n

H X M X K M n X B M

G G

G

In interpreting this one must rememb er that the right side concerns homotopy

n

classes of genuine natural transformations X B M and not just natural

transformations in the homotopy category The latter would b e directly com

putable in terms of nonequivariant comology

Example If M is a contravariant functor from hG to not necessarily

Ab elian groups then we can regard B M as a G space and so obtain an Eilenb erg

Mac Lane Gspace K M B M

Example A setvalued functor M on hG is the same thing as a continu

ous setvalued functor on G Applying to such an M we obtain an Eilenb erg

H

Mac Lane Gspace K M Its xed p oint spaces K M are homotopy equiv

alent to the discrete spaces M GH but the Gspace K M generally has

nontrivial cohomology groups in arbitrarily high dimension For setvalued co ef

cient systems M and M let N at M M b e the set of natural transformations

G

M M Then Theorem and the discreteness of M give isomorphisms

X K M X M N at X M

G G G

This may seem frivolous at rst sight but in fact the spaces K M are cen

tral to equivariant homotopy theory For example we shall see later that the

isomorphisms just given sp ecialize to give a classication theorem for equivariant

and to reprove the classical classication of nonequivariant bundles bundles

The relevant K M s are sp ecial cases of those in the following basic denition

EILENBERGMAC LANE GSPACES AND UNIVERSAL F SPACES

Definition A family F in G is a set of subgroups of G that is closed

under sub conjugacy if H F and g K g H then K F An F space is a

hG S ets Gspace all of whose isotropy groups are in F Dene a functor F

by sending GH to the p oint set if H F and to the empty set if H F

Dene the universal F space E F to b e F It is universal in the sense that for

an F space X of the homotopy typ e of a GCW complex there is one and up to

homotopy only one Gmap X E F Dene the classifying space of the family

F to b e the orbit space B F E F G

In thinking ab out this example it should b e rememb ered that there are no

maps from a nonempty set to the empty set In particular there are no Gmaps

only X E F if X is not an F space This also shows that the functor F

makes sense if the given set F of subgroups of G is a family We augment the

denition with the following relative version It will b ecome very imp ortant later

Definition For a subfamily F of a family F dene E F F to b e the

Let A cob er of the based Gmap unique up to homotopy E F E F

b e the family of all subgroups of G and let E F E A F Since E A is

Gcontractible E F is equivalent to the unreduced susp ension of E F with one

H

of the cone p oints as basep oint The space E F is contractible if H F and

is the twop oint space S if H F For F F the Gspace E F F is

equivalent to E F E F

The following observation will b ecome valuable when we examine the structure

of equivariant classifying spaces

Lemma Let F b e a family in G and H b e a subgroup of G

a Regarded as an H space E F is E F jH where

F jH fK jK F and K H g

H H

b If H F then regarded as a WH space E F is E F where

H

F fLjL K H for some K F such that H K NH g

The classical example is F feg An fegspace X is a Gspace all of whose

isotropy groups are trivial That is X is a free Gspace Then EG E feg is

exactly the standard example of a free contractible Gspace and the quotient map

EG BG is a principal Gbundle Given the result that pullbacks of bundles

along homotopic maps are homotopic we have already proven that is universal

V CATEGORICAL CONSTRUCTIONS EQUIVARIANT APPLICATIONS

Indeed if p E B is a principal Gbundle we have a unique homotopy class

of Gmaps f E EG The map f B BG that is obtained by passage to

orbits from f is the classifying map of p Certainly p is equivalent to the bundle

obtained by pulling back along f

When G is discrete the ordinary homology and cohomology of the Gspaces

E F admit descriptions as Ext groups generalizing the classical identication of

the homology and cohomology of groups with the homology and cohomology of

K s This can b e seen from the pro jectivity of the cellular chains C E F

and insp ection of denitions or by collapse of the universal co ecients sp ectral

sequences Write ZF for the free Ab elian group functor comp osed with the

functor F

Proposition Let G b e discrete For a covariant co ecient system N and

a contravariant co ecient system M

G G

ZF ZF E F M Ext E F N Tor N and H M H

G G

CHAPTER VI

Diagrams The Homotopy Theory of

by Rob ert J Piacenza

Elementary homotopy theory of diagrams

A substantial p ortion of the homotopy homology and cohomology theory of

Gspaces X dep ends only on the underlying diagram of xed p oint spaces X

G U There is a vast and growing literature in which the homotopy theory of

spaces is generalized to a homotopy theory of diagrams of spaces that are indexed

on arbitrary small indexing categories The purp ose of this chapter is to outline

this theory and to demonstrate the connection b etween diagrams and equivariant

theory A very partial list of sources for further reading is given at the end of this

section

Throughout the chapter we let U b e the cartesian category of compactly gener

ated weak Hausdor spaces and let J b e a small top ological category over U with

J

discrete ob ject space Dene U to b e the category of continuous contravariant

U valued functors on J Its ob jects are called either diagrams or J spaces its

J

morphisms which are natural transformations are called J maps Note that U

is a top ological category its hom sets are spaces and comp osition is continuous

Let I b e the unit interval in U If X and Y are diagrams then a homotopy

from X to Y is a J map H I X Y where I X is the diagram dened

on ob jects j jJ j by I X j I X j and similarly for morphisms of J In

the usual way homotopy denes an equivalence relation on the J maps that gives

J

rise to the quotient homotopy category hU We denote the homotopy classes of

J

J maps from X to Y by hU X Y abbreviated hX Y An isomorphism in

VI THE HOMOTOPY THEORY OF DIAGRAMS

J

hU will b e called a homotopy equivalence

A J map is called a J cobration if it has the J homotopy extension prop erty

abbreviated J HEP The basic facts ab out cobrations in U apply readily to

J cobrations

J

The following standard results for spaces are inherited by the category U

ariance of pushouts Supp ose given a commutative di Theorem Inv

agram

f

P P

B A

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P i

P

P

P

P

P

P

P

P

P

P P

P

Y X

P A B

P

P

0

P

P

f

P

P

0 P

P

P

P

i

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

X Y

in which i and i are J cobrations f and f are arbitrary J maps and

are homotopy equivalences and the front and back faces are pushouts Then the

induced map on pushouts is also a homotopy equivalence

Theorem Invariance of colimits over cofibra tions Supp ose

given a homotopy commutative diagram

i

i i

k

0 1

k

X X

X

k 1 0

f f

f

j j j

1 0

k

k

Y Y Y

J k

in U where the i and j are J cobrations and the f are homotopy equivalences

k k

k k k

Then the map colim f colim X colim Y is a homotopy equivalence

k k k

The reader will readily accept that other such standard results in the homotopy

theory of spaces carry over directly to the homotopy theory of diagrams

W G Dwyer and D M Kan An obstruction theory for diagrams of simplicial sets Pro c Kon

Ned Akad van Wetensch AInd Math

W G Dwyer and D M Kan Singular functors and realization functors Pro c Kon Ned

Akad van Wetensch AInd Math

W G Dwyer K M Kan and J H Smith Homotopy commutative diagrams and their real

izations J Pure and Appl Alg

E Dror Farjoun Homotopy and homology of diagrams of spaces Springer Lecture Notes in

Mathematics Vol

HOMOTOPY GROUPS

E Dror Farjoun Homotopy theories for diagrams of spaces Pro c Amer Math So c

A Heller Homotopy in functor categories Trans Amer Math So c

R J Piacenza Homotopy theory of diagrams and CWcomplexes over a category Canadian J

Math

K Sarnowski Homology and cohomology of diagrams of top ological spaces Thesis University

of Alaska

Y Shitanda Abstract homotopy theory and homotopy theory of functor category Hiroshima

Math J

I Mo erdijk and J A Svensson A Shapiro lemma for diagrams of spaces with appliations to

equivariant top ology Comp ositio Mathematica

Homotopy Groups

n n

Let I b e the top ological ncub e and I its b oundary For an ob ject j jJ j

J

U denote the asso ciated represented functor its value on an ob ject k is let j

J

the space U k j

J

Definition By a pair X Y in U we mean a J space X together with

a sub J space Y Morphisms of pairs are dened in the obvious way Similar

J

denitions apply to triples nads etc Let j Y b e a morphism in U By

oneda lemma is completely determined by the p oint id y Y j the Y

j

For each n dene

j n n

X Y hI I fg j X Y Y

n

where y id Y j serves as a basep oint and all homotopies are homotopies

j

of triples relative to The reader may formulate a similar denition for the

j

absolute case X For n we adopt the convention that I f g and

n

I fg and pro ceed as ab ove These constructions extend to covariant functors

J j

on U From now on we shall often drop from the notation X Y

n

The following prop osition follows immediately from the Yoneda lemma

j

X j and Proposition There are natural isomorphisms X

n

n

j

X j Y j that preserve the group structures when n in X Y

n

n

the absolute case the relative case requires n

As a direct consequence of Prop osition we obtain the usual long exact se quences

VI THE HOMOTOPY THEORY OF DIAGRAMS

Proposition For X Y and j as in Denition there exist natural

b oundary maps and long exact sequences

j j j

j j

Y X Y X X Y

n

n n

j

of groups up to Y and p ointed sets thereafter

J

Definition A map e X Y X Y of pairs in U is said to b e

an nequivalence if ej X j Y j X j Y j is an nequivalence in U

for each j jJ j A map e is said to b e a weak equivalence if it is an nequivalence

for each n Observe that e is an nequivalence if for every j jJ j and

j j

j Y e X Y X Y e is an isomorphism for p n

p p

and an epimorphism for p n The reader may easily formulate similar denitions

for J maps e X X the absolute case

Cellular Theory

In this section we adapt Mays preferred approach to the classical theory of CW

complexes to develop a theory of J CW complexes

n n

b e the top ological n disk and S the top ological nsphere Of Let D

n n

course these spaces are homeomorphic to I and I resp ectively We shall

n

construct cell complexes over J by the pro cess of attaching cells of the form D

n

j by attaching morphisms with domain S j

J

Definition A J complex is an ob ject X of U with a decomp osition

p

X where X colim

p

a

n

j X D

A

0

and inductively

a

p p n

X X j D

f A

p

n p

S j for some attaching J map f X here for each p

A

p

fj j A g is a set of ob jects of J We call X a J CW complex if X is a

p

J complex such that n p for all p and A

p

Now J sub complexes and relative J complexes are dened in the obvious way

We adopt the standard terminology for CWcomplexes for J CWcomplexes with

out further comment

The following technical lemma reduces directly to its space level analog

CELLULAR THEORY

Lemma Supp ose that e Y Z is an nequivalence Then we can

J

complete the following diagram in U

i i

0 1

n n n

o

S j S I j S j

s

v

s

g

v

h

s

v

s

v

s

v

s

s v

s

v

s

v

s

v

y s

e

o

i

Z Y

eL e

dH d

v

L

v H

v

L

f g

h

v H

L

v

v H

L

v

v H

L

v

n n n

o

D j D I j D j

i i

1 0

to obtain the following results From here we pro ceed exactly as in Ix

Theorem J HELP If X A is a relative J CW complex of dimension

n and e Y Z is an nequivalence then we can complete the following

J

diagram in U

o

A I

A A

x

x

x

x

x

x

x

x

x

e

o

Z Y

ccF

A

F

A

F

A

F

F

A

o

X I

X X

Theorem Whitehead Let e Y Z b e an nequivalence and X b e

a J CW complex Then e hX Y hX Z is a bijection if X has dimension

less than n and a surjection if X has dimension n If e is a weak equivalence then

e hX Y hX Z is a bijection for all X

Cor ollary If e Y Z is an nequivalence b etween J CW complexes

of dimension less than n then e is a J homotopy equivalence If e is a weak

equivalence b etween J CW complexes then e is a J homotopy equivalence

oximation Let X A and Y B b e rela Theorem Cellular Appr

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

J

Y B b e a map of pairs in U whose restriction to X A is cellular Then f is

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

Corollary Let X and Y b e J CW complexes Then any J map f

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

VI THE HOMOTOPY THEORY OF DIAGRAMS

Next we discuss the lo cal prop erties of J CW complexes First we develop some

preliminary concepts Let X b e a J space and for each j jJ j let t X j

j

colim X b e the natural map of X j into the colimit Observe that for each

J

morphism s i j of J t t X s For each subspace A colim X

j i J

we dene Aj t A for each morphism s i j of J we dene As

j

Aj Ai to b e the restriction of X s As usual we apply the k ication

functor to ensure that all spaces dened ab ove are compactly generated One

quickly checks that A is a J space that colim A A and that there is a natural

J

inclusion A X To simplify notation we write

XJ colim X

J

from now on

Definition A pair X A is a J neighb orho o d retract pair abbreviated

of XJ such that A U and a J NR pair if there exists an op en subset U

retraction r U A A pair X A is a J neighb orho o d deformation retract

pair abbreviated J NDR pair if X A is a J NR pair and the J map r is a

J deformation retraction

de Let X b e a J CW complex The functor colim sends the J space A j

J

termined by a space A and ob ject j to the space A and it preserves colimits

Therefore the cellular decomp osition of X determines a natural structure of a CW

complex on XJ its attaching maps are the images under the functor colim of

J

the attaching J maps of X One may also check that if A is a sub complex of XJ

p

then A has a natural structure of a sub complex of X In particular if A is the

p p

pskeleton of XJ then A X is the pskeleton of X

Proposition Local contractibility Let X b e a J CW complex and

and A fag b e a p oint of XJ Then there is an ob ject j jJ j such that A j

X A is a J NDR pair

Proof Let a b e in the pskeleton but not in the p skeleton of XJ Then

p p

there is a unique attaching map f S j X such that a is in the interior

p

To construct the required neighb orho o d U rst take j of D It follows that A

an op en ball U contained in the interior of B and centered at a Then U is a

p

p

neighb orho o d in XJ that contracts to A One then extends U inductively cell

by cell by the usual space level pro cedure to construct the required neighb orho o d

U

THE HOMOLOGY AND COHOMOLOGY THEORY OF DIAGRAMS

Proposition Let X b e a J CW complex and A b e a sub complex of XJ

Then X A is a J NDR pair

Proof It follows from J HELP that A X is a J cobration Just as on

the space level a J cobration is the inclusion of a J NDR pair

R J Piacenza Homotopy theory of diagrams and CWcomplexes over a category Canadian J

Math

The homology and cohomology theory of diagrams

The ordinary homology and cohomology theories of Ix are sp ecial cases of a

J

construction that applies to the category U for any J The dierence is that

the theory in Ix started with GCW complexes and then passed to the asso ciated

diagrams dened on the orbit category of G whereas we here exploit the theory of

J CW complexes There is again a vast literature on the cohomology of diagrams

some relevant references b eing listed in Section

Dene a J co ecient system to b e a continuous contravariant functor M J

A b Continuity ensures that M factors through the homotopy category hJ Let

hJ

A b b e the category of J co ecient systems It is an Ab elian category and

we can do homological algebra in it As in Ix a covariant homotopy invariant

functor U A b induces a functor from J spaces to J co ecient systems by

comp osition we name such functors by underlining the name of the given functor

Of course we also have the notion of a covariant J co ecient system

n

Let X A b e a relative J CW complex with n skeleton X and observe that

a a

n n n n n

S j S j D j X X

j j

where the indicates the addition of disjoint basep oints Dene a chain complex

J

C X A in A b called the J cellular chains of X A by setting

n n

X A H X X Z C

n n

n n n

The connecting homomorphisms of the triples X X X sp ecify the dif

ferential

X A C X A d C

n n

VI THE HOMOTOPY THEORY OF DIAGRAMS

Clearly implies that

X

X Aj J j j Z H C

n

j

The construction is functorial with resp ect to cellular maps X A Y B

For a covariant J co ecient system N dene the cellular chain complex of

X A with co ecients N by

C X A N C X A N

J

where the tensor pro duct on the right is interpreted as the co end over J Passing

to homology we obtain the cellular homology H X A N

For a contravariant J co ecient system M dene the cellular co chain complex

of X A with co ecients M by

C X A M Hom C X A M

J

Passing to cohomology we obtain the cellular cohomology H X A M

Theorem Cellular homology and cohomology for pairs of J CW complexes

satisfy the standard Eilenb ergSteenro d axioms suitably reformulated for dia

grams

Remark We may extend the cellular theory to arbitrary pairs of diagrams

by means of cellular approximations see Prop osition That is we extend our

homology and cohomology theories to theories that carry weak equivalences to

isomorphisms We may also adapt Ilmans construction of equivariant singular

theory to construct a singular theory for diagrams Of course the singular theory

is isomorphic to the cellular theory on the category of J CW complexes

S Illman Equivariant singular homology and cohomology Memoirs Amer Math So c No

J

The closed mo del structure on U

Just as the category of spaces has a closed mo del structure in the sense of

Quillen so do es the category of Gspaces for any G This p oint of view has not

b een taken earlier since the conclusions are obvious to the exp erts and p erhaps

not very helpful to the novice on a rst reading However since the homotopical

prop erties of categories of diagrams are likely to b e less familiar than those of the

category of spaces it is valuable to understand how they inherit mo del structures

J

THE CLOSED MODEL STRUCTURE ON U

from the standard mo del structure on U which is the sp ecial case of the trivial

category J in the denitions here We use the name q bration and q cobration

for the mo del structure brations and cobrations to avoid confusion with other

kinds of brations and cobrations The weak equivalences of the mo del structure

will b e the weak equivalences that we have already dened an acyclic q bration

is one that is a weak equivalence and similarly for acyclic q cobrations Consider

diagrams

A X

g

f

B Y

The map g has the left lifting prop erty LLP with resp ect to f if one can always

ll in the dotted arrow The right lifting prop erty RLP is dened dually

Definition A J map f X Y is a q bration if f j Y j X j

is a Serre bration for each ob ject j jJ j Observe that f is a q bration if f has

n

A map g A B the homotopy lifting prop erty for all ob jects of the form I j

is a q cobration if it has the LLP with resp ect to all acyclic q cobrations

J

Theorem With the structure just dened U is a mo del category

Proof Just as as for spaces one quickly checks Quillens axioms using the

factorization lemma b elow to verify the factorization axiom M

As for spaces the pro of leads directly to the following characterizations of q

cobrations and of acyclic q brations

Corollary A J map g A B is a q cobration if and only if it is a

retract of the inclusion A B of a relative J complex B A

Corollary A J map f X Y is an acyclic q bration if and only if

n n

D j it has the RLP with resp ect to each q cobration S j

factorization lemma Any J map f X Y Lemma Quillens

can b e factored as f p g where g is a q cobration and p is an acyclic q bration

VI THE HOMOTOPY THEORY OF DIAGRAMS

Proof We construct a diagram

g g

1 0

X

Z Z

A

A

A

A

A

p

0

A

A

A

f

Y

n

as follows Let Z X and p f Having obtained Z consider the set of

all diagrams of the form

t

q

n

S j

Z

p

n 1

q

D j

Y

s

Forming the copro duct over all of the left vertical arrows we may dene g

n

n n

Z Z by the pushout diagram

t

q

n

S j

Z

p

n1

q

n

D j

Z

s

We have allowed the zero dimensional pair D S fptg in this construc

n

tion Dene p Z Y by pushing out along p and the copro duct of the

n n

maps s Then let

n

Z colim Z p colim p and g colim g g g

n n n

One may check that g has the LLP with resp ect to each acyclic q bration and

n

by the small ob ject argument based on the compactness of the D that p is an

acyclic q bration

J J

Let hU b e the lo calization of hU obtained by formally inverting the weak

J

equivalences The mo del structure implies that hU is equivalent to the homotopy

category of J CW complexes as we indicate next

Lemma Let X colim X taken over a sequence of J cobrations such that

n

each X has the homotopy typ e of a J CW complex Then X has the homotopy

n

typ e of a J CW complex

ANOTHER PROOF OF ELMENDORFS THEOREM

Proof Up to homotopy we may approximate the sequence by a sequence of

J CW complexes and cellular inclusions we then use the homotopy invariance of

colimits Theorem

The following prop osition follows easily

Proposition Each J complex is of the homotopy typ e of a J CW com

plex

J

Theorem Approximation theorem There is a functor U

J J

U and a natural transformation id such that for each X U X is

a J complex and X X is an acyclic q bration

Proof Applying Lemma to the inclusion of the empty set in X we obtain

an acyclic q bration X X By the explicit construction we see that X

is a J complex is a functor and is a natural transformation

The following corollary is immediate from the previous two results

J

Corollary The category hU is equivalent to the homotopy category of

J CW complexes

D G Quillen Homotopical Algebra Springer Lecture Notes in Mathematics Vol

o ok of W G Dwyer and J Spalinsky Homotopy theories and mo del categories In Handb

edited by IM James North Holland pp Algebraic Top ology

Another pro of of Elmendorf s theorem

The theory of diagrams leads to an alternative pro of of Elmendorf s theorem

V one which gives a precise cellular p ersp ective and illustrates the force of

mo del category techniques We adopt the notations of Vx

Observe that the xed p oint diagram functor from Gspaces to G spaces

for a space X regarded as a Gtrivial Gspace carries X GH to X GH

Thus it preserves cells It also preserves the pushouts relevant to cellular theory

Lemma If

A X

i

B Y

VI THE HOMOTOPY THEORY OF DIAGRAMS

is a pushout of Gspaces in which i is a closed inclusion then

X A

i

B Y

is a pushout of G spaces

Proof Stripping away the top ology we see that this holds on the set level

since every Gset is a copro duct of orbits One may then check that the top ologies

agree

Theorem Each G complex or G CW complex Y GU is isomorphic to

X for some Gcomplex or GCW complex X Therefore is an isomorphism

b etween the category of Gcomplexes or G CW complexes and the category of

G complexes or G CW complexes

Proof The functor carries Gcomplexes to G complexes since it preserves

cells the relevant pushouts and ascending unions The assertion follows since

is full and faithful inductively the attaching maps of Y are in the image of

This leads to our alternative version of V

Theorem Elmendorf There is a functor GU GU and a

natural transformation id such that X is a Gcomplex X is a

G complex and X X is a weak equivalence of Gspaces for each G

space X Therefore and induce an equivalence of categories b etween hGU

and hGU

Proof We construct the functor and transformation by using the functor

and transformation p given in Theorem on the level of diagrams and using

Theorem to transp ort from G complexes to Gcomplexes The result follows

from the cited results and Corollary

Corollary Let Y b e a Gspace of the homotopy typ e of a GCW complex

Then for any G space X

hGU Y X hY X hGU Y X

Proof This follows from Theorem and generalities ab out mo del cate gories

ANOTHER PROOF OF ELMENDORFS THEOREM

In turn this implies the following comparison with the original form V of

Elmendorf s theorem

Corollary Write and for the constructions given in V For a

G space X there is a weak equivalence of Gspaces X X such that

is natural up to homotopy and the following diagram commutes up to homotopy

X

X

G

G

w

w

G

w

G

w

G

w

G

w

G

0 w

G

w

w X

VI THE HOMOTOPY THEORY OF DIAGRAMS

CHAPTER VI I

theory and Classifying Spaces Equivariant Bundle

The denition of equivariant bundles

Equivariant bundle theory can b e develop ed at various levels of generality We

of a compact Lie group We set G and we assume given a subgroup

let q G b e the quotient homomorphism That is we consider an extension

of compact Lie groups

G

Many sources restrict attention to split extensions but we see little p oint in that

By far the most interesting case is G When is O n or U n this case

will lead to real and complex equivariant K theory

Dene a principal bundle to b e the pro jection to orbits p E E

B of a free space E Note that G acts on the base space B Let F b e a

space By a Gbundle with structural group total group and b er F we

F B induced by a principal bundle E E is mean the pro jection E

called the asso ciated principal bundle Although we prefer to think of bundles this

way it is not hard to give an intrinsic characterization of when a Gmap Y B

that is a bundle with b er F is such a bundle

When G we shall refer to G bundles rather than to G

bundles Here it is usual to require the b er F b e a space A principal G

bundle E has actions by G and that commute with one another it is usual to

write the action of on the right and the action of G on the left Equivariant

n

vector bundles t into this framework a G O nbundle with b er R is called

an nplane Gbundle and similarly in the complex case The tangent and normal

VI I EQUIVARIANT BUNDLE THEORY AND CLASSIFYING SPACES

bundles of a smo oth Gmanifold give examples

Example A nite Gcover p Y B is a Gmap that is also a nite

cover Such a map is necessarily a G bundle with b er the set F

n n

f ng Its asso ciated principal G bundle E is the subspace of MapF Y

n

consisting of the bijections onto b ers of p

Classical bundle theory readily generalizes to the equivariant context and we

content ourselves with a very brief summary of some of the main p oints A prin

cipal bundle is said to b e trivial if it is equivalent to a bundle of the form

q id U G U

H

e q maps isomorphically onto H and U where H G

is an H space regarded as a space by pullback along q Provided that E and

therefore also B are completely regular a principal bundle p E B is

lo cally trivial If in addition B is paracompact then p is numerable Numerable

bundles satisfy the equivariant bundle covering homotopy prop erty and

a numerable bundle E over B I is equivalent to the bundle E fg I

Therefore the pullbacks of a numerable bundle along homotopic Gmaps

are equivalent

R K Lashof Equivariant bundles Ill J Math

R K Lashof and J P May Generalized equivariant bundles Bulletin de la So ci ete Mathema

tique de Belgique

L G Lewis Jr J P May and M Steinb erger with contributions by J E McClure Equiv

ariant stable homotopy theory IVx Springer Lecture Notes in Mathematics Vol

The classication of equivariant bundles

X b e the set of equivalence classes of principal bundles Let B

with base Gspace X We assume that X has the homotopy typ e of a GCW

complex and we check that this implies that any bundle over X has the homotopy

typ e of a CW complex Then Elmendorf s theorem V sp ecializes to give a

classication theorem for principal bundles

Definition Dene F to b e the family of subgroups of such

that e and observe that an F space is the same thing as a free

space Write

E E F and B E

THE CLASSIFICATION OF EQUIVARIANT BUNDLES

and let

E B

b e the resulting principal bundle In the case G write F

G

F G

E E G and B B G

G G

Observe that since E is a contractible space B is a mo del for B

that carries a particular action by G

Theorem The bundle E B is universal That is

pullback of along Gmaps X B gives a bijection

X B B X

G

It is crucial to the utility of this result to understand the xed p oint structure

of B For any principal bundle p E B and any H G one

H

can check that B is the disjoint union of the spaces pE where runs over

the conjugacy classes of subgroups such that e and q H

Dene

N Z

where Z is the centralizer of in the equality here is an easy observation

Then E is a principal W bundle and pE is its base space We can go

H

on to analyze the structure of B as a W H space In the case of the universal

G

bundle we can determine the structure of E by use of IV Putting things

together we arrive at the following conclusion

Theorem For a subgroup H of G

a

H

B B

where the union runs over the conjugacy classes of subgroups of such that

e and q H as a W H space

G

a

H

B W H B W

G

V

where the union runs over the q N H conjugacy classes of such groups and

G

V W is the image of W in W H

G

VI I EQUIVARIANT BUNDLE THEORY AND CLASSIFYING SPACES

Here by use of Lie group theory V has nite index in W H

G

Sp ecializing to G we see that the subgroups of such that e

are exactly the twisted diagonal subgroups

fh hjh H g

where H is a subgroup of G and H is a homomorphism Let N

N and observe that

G

N fg jg N H and h g hg for all h H g

G

Therefore N coincides with the centralizer

f j h h for all h H g

Let

W W and V W W H

G G

As usual let RepG denote the set of conjugacy classes of homomorphisms

G Dene an action of the group W H on the set RepH by letting

G

g H b e the conjugacy class of g where for g N H g H is

G

the homomorphism sp ecied by g h g hg Observe that the isotropy

group of is V

Theorem For a subgroup H of G

a

H

B B

G

where the union runs over RepH as a W H space

G

a

H

B W H B W

G G V

where the union runs over the orbit set RepH W H

G

It is imp ortant to observe that the group W need not split as a pro duct

V in general Therefore in order to fully understand the classifying G

spaces for G bundles one is forced to study the classifying spaces for the

more general kind of bundles that we have intro duced These are complicated

ob jects and their study is in a primitive state In particular rather little is

known ab out equivariant characteristic classes Such classes are understo o d in

Borel cohomology however By the universal prop erty of E there is a

map E E which is unique up to homotopy The induced Gmap

E B is a nonequivariant equivalence and so induces an isomorphism

SOME EXAMPLES OF CLASSIFYING SPACES

on Borel cohomology The pro jection EG E E is clearly a homotopy

equivalence and it induces an equivalence

EG E EG E E B

G

This already implies the following calculation We again denote Borel cohomology

by H for the moment

G

H B With eld co Theorem With any co ecients H B

G

H BG H B as an H BGmo dule ecients H B

G

G

The interpretation is that the Borel cohomology characteristic classes of a prin

cipal G bundle E over X are determined by the H BGmo dule structure

X together with the nonequivariant characteristic classes of the bundle on H

G

EG E over EG X

G G

We shall later see that generalized versions of the AtiyahSegal completion the

orem and of the Segal conjecture give calculations of the characteristic classes of

G bundles in equivariant K theory and in equivariant cohomotopy

R K Lashof and J P May Generalized equivariant bundles Bulletin de la So ciete Mathema

tique de Belgique

J P May Characteristic classes in Borel cohomology J Pure and Applied Algebra

Some examples of classifying spaces

It is often valuable to have alternative descriptions of universal bundles We

have Grassmannian mo dels when is an orthogonal or unitary group These lead

to go o d mo dels for the classifying spaces for equivariant K theory and just as

nonequivariantly they are useful for the pro of of equivariant versions of the Thom

cob ordism theorem

Example For a real inner pro duct Gspace V let BO n V b e the Gspace

of nplanes in V and let EO n V b e the Gspace whose p oints are pairs consisting

of an nplane in V and a vector v The map EO n V BO n V that

sends v to is a real nplane Gbundle Provided that V is large enough say

the direct sum of innitely many copies of each irreducible real representation of

G p is a universal real nplane Gbundle A similar construction works in the complex case

VI I EQUIVARIANT BUNDLE THEORY AND CLASSIFYING SPACES

Clearly a principal bundle E is universal if and only if E is contractible

for F Using the fact that the space of Gmaps from a free GCW

complex to a nonequivariantly contractible Gspace is contractible one can use this

criterion to obtain a simple mo del that has particularly go o d naturality prop erties

Regard EG as a space via q G and dene

SecE G E Map E G E

to b e the sub space consisting of those maps f EG E such that the

comp osite of E q E EG and f is the identity map Note that

SecE G E G MapE G E

since E G is homeomorphic to EG E

Theorem The space SecE G E is a universal principal bundle

and therefore the Gspace SecE G E is a mo del for B In particular

the G space MapE G E is a universal principal G bundle and therefore

the Gspace Map E G E is a mo del for B

G

Since we are interested in maps from GCW complexes into classifying spaces

the fact that these mo dels need not have the homotopy typ es of GCW complexes

need not concern us

Observe that the map E B induces a natural Gmap

B SecE G E SecE G B

where SecE G B is the Gspace of maps f EG B such that the comp osite

of f and B q B BG is EG BG With G this map is

B MapE G B

G

These maps have bundle theoretic interpretations Restricting for simplicity to

the case G let

X B B X

G G G

b e the set of equivalence classes of G bundles over X and let B X b e

the set of equivalence classes of nonequivariant bundles over X By adjunction

a Gmap X Map E G B is the same as a map EG X B Thus the G

SOME EXAMPLES OF CLASSIFYING SPACES

represented equivalent of is the Borel construction on bundles that was relevant

to Theorem it gives

B X B EG X

G G

It is imp ortant to know how much information this construction loses hence it is

imp ortant to know how near is to b eing an equivalence Elementary covering

space theory gives the following result

Proposition If is discrete then the Gmap of is a homeomor

phism If but not necessarily G is discrete then the Gmap of is a

homeomorphism

An Ab elian compact Lie group is the pro duct of a nite Ab elian group and

a torus Using ordinary cohomology to study the nite factor and continuous

cohomology to handle the torus factor Lashof May and Segal proved another

result along these lines

Theorem If G is a compact Lie group and is an Ab elian compact Lie

group then the Gmap B MapE G B is a weak equivalence

G

Consequences of the Sullivan conjecture will tell us much more ab out these

maps To see this we will need to know the b ehavior of the maps on xed p oint

H

spaces We have determined the xed p oint spaces B and it is clear that

H

SecE G B SecB H B

is the space of maps f BH B such that

B q f B i BH B G

where i H G is the inclusion and we take B i to b e the quotient map

E GH E GG In particular

SecB H B G B MapB H B

Lemma Let satisfy e and q H Dene a homomor

H by q and observe that q i phism

H G The restriction of

H H

B SecB H B

VI I EQUIVARIANT BUNDLE THEORY AND CLASSIFYING SPACES

to B is the adjoint of the classifying map

B BH B B H B

Therefore if G the restriction of

H H

B MapB H B

G

to B H is the adjoint of the map of classifying spaces

BH B B H B B

where H is dened by h h

Consider what happ ens on comp onents In nonequivariant homotopy theory

maps b etween the classifying spaces of compact Lie groups have b een studied for

many years One fo cus has b een the question of when passage to classifying maps

B RepG B G B

is a bijection We now see that for H G a map BH B not in the

image of B corresp onds to a principal bundle over BH that do es not come

from a principal G bundle over an orbit GH The equivariant results ab ove

imply that there are no such exotic maps if is either nite or Ab elian The

Sullivan conjecture will give information ab out general compact Lie groups

under restrictions on G

R K Lashof J P May and G B Segal Equivariant bundles with Ab elian structural group

Contemp orary Math Vol

J P May Some remarks on equivariant bundles and classifying spaces Asterisque

CHAPTER VI I I

The Sullivan Conjecture

Statements of versions of the Sullivan conjecture

We dened the homotopy orbit space of a Gspace X to b e

X EG X

hG G

and we dened the homotopy xed p oint space of X dually

hG G

X MapE G X Map E G X

G

is the space of Gmaps EG X The pro jection EG induces

G G hG

X Map X MapE G X X

It sends a xed p oint to the constant map EG X at that xed p oint It is very

natural to ask how close this map is to b eing a homotopy equivalence Thinking

equivariantly it is even more natural to ask how close the Gmap

X Map X MapE G X

is to b eing a Ghomotopy equivalence Since a Gmap f X Y that is a

nonequivariant equivalence induces a weak equivalence of Gspaces

MapW Y Map W X

for any free GCW complex W such as EG one cannot exp ect to b e an equiva

lence in general Very little is known ab out this question for general nite groups

However for nite pgroups G to which we restrict ourselves unless we sp ecify

otherwise the Sullivan conjecture gives a b eautiful answer We agree to work in

VI I I THE SULLIVAN CONJECTURE

the categories hU and hGU implicitly applying CW approximation This allows

us to ignore the distinction b etween weak and genuine equivalences

Sullivan conjecture Let X b e a nilp otent Theorem Generalized

nite GCW complex Then the natural Gmap

X MapE G X

p p

is an equivalence

The hyp othesis that X b e nilp otent can b e removed by applying the Bouseld

Kan simplicial completion on xed p oint spaces and then assembling these com

pleted xed p oint spaces to a global Gcompletion by means of Elmendorf s con

struction This equivariant interpretation of the Sullivan conjecture was noticed by

Haeb erly who also gave some information for nite groups that are not pgroups

Lo oking at xed p oints under H G and noting that EG is a mo del for EH we

see that the result immediately reduces to the xed p oint space level

Theorem Miller Carlsson Lannes Let X b e a nilp otent nite G

CW complex where G is a nite pgroup Then the natural map

G G hG G

X Map E G X X X

p p p p

is an equivalence

Again the nilp otence hyp othesis is unnecessary provided that one understands

X to mean the BouseldKan completion of X which generalizes the nilp otent

p

G G

completion that we dened and takes X and not X as the source there

p p

is a natural map

G G

X X

p p

but it is not an equivalence in general When G acts trivially on X the result was

rst proven by Miller and he deduced the following p owerful consequence

Theorem Miller Let G b e a discrete group such that all of its nitely

generated subgroups are nite and let X b e a connected nite dimensional CW

complex Then F B G X

To deduce this from Theorem one rst observes that any map BG X

induces the trivial map of fundamental groups and so lifts to the universal cover

n

while a map BG X for n trivially lifts to the universal cover Thus

one can assume that X is simply connected Note that this reduction dep ends

STATEMENTS OF VERSIONS OF THE SULLIVAN CONJECTURE

on the fact that we are here working with nite dimensional and not just nite

complexes and one must generalize Theorem accordingly this seems to require

trivial action on X One then applies an inductive argument to reduce to the

case G Zp Here the weak equivalence X MapB G X implies that

p p

F B G X and this implies that F B G X

p

The general case of Theorem reduces immediately to the case when G Zp

by induction on the order of G To see this consider an extension

C G J

hC hJ hG

where C is cyclic of order p For any Gspace Y Y is equivalent to Y In

fact by passing to Gxed p oints by rst passing to C xed p oints and then to

J xed p oints we obtain a homeomorphism

G C J

MapEJ E G Y Map EJ MapE G Y

Since EJ EG is a free contractible Gspace and EG is a free contractible C space

this gives the stated equivalence of homotopy xed p oint spaces The equivalence

C hC

X X is a J map hence it induces an equivalence on passage to

p p

J homotopy xed p oint spaces and the map of Theorem coincides with the

comp osite equivalence

hG G C J C hJ hC hJ

X X X X X

p p p p p

When G Zp Theorem was proven indep endently by Lannes and Miller

using nonequivariant techniques and by Carlsson using equivariant techniques

Lannes later gave a variant of his original pro of that generalizes the result uses

equivariant ideas and enjoys a pleasant conceptual relationship to Smith theory

We shall sketch that pro of in the following three sections

There is a basic principle in equivariant top ology to the eect that when working

at a prime p results that hold for pgroups can b e generalized to ptoral groups

G which are extensions of the form

T G

The p oint is that the circle group can b e approximated by the union of its

n

psubgroups of p th ro ots of unity and an r torus T can b e approximated by

n

r

of its psubgroups It is not hard to see that the map the union

n n

B BT induces an isomorphism on mo d p homology Using this basic idea

Notb ohm generalized Theorem to ptoral groups

VI I I THE SULLIVAN CONJECTURE

Theorem Notbohm The generalized Sullivan conjecture Theorem

remains true as stated when G is a ptoral group

Technically this still works using BouseldKan completion for pgo o d G

spaces X for which X X is a mo d p equivalence

p

A K Bouseld and D M Kan Homotopy limits completions and lo calizations Springer

Lecture Notes in Mathematics Vol

G Carlsson Equivariant stable homotopy theory and Sullivans conjecture Invent Math

W Dwyer Haynes Miller and J Neisendorfer Fibrewise completion and unstable Adams

sp ectral sequences Israel J Math

JP Haeb erly Some remarks on the Segal and Sullivan conjectures Amer J Math

des pgroup es abelien elementaires in Homotopy J Lannes Sur la cohomologie mo dulo p

theory Pro c Durham Symp London Math So c Lecture Notes

H R Miller The Sullivan conjecture on maps from classifying spaces Annals of Math

and corrigendum Annals of Math

H R Miller The Sullivan conjecture and homotopical representation theory Pro c Int Cong

of Math Berkeley Ca USA

D Notb ohm The xedp oint conjecture for ptoral groups Springer Lecture Notes in Mathe

matics Vol

L Schwartz Unstable mo dules over the Steenro d algebra and Sullivans xed p oint conjecture

University of Chicago Press

Algebraic preliminaries Lannes functors T and Fix

Let V b e an elementary Ab elian pgroup xed throughout this section and the

next It would suce to restrict attention to V Zp The notation V indicates

that we think of V ambiguously as b oth a vector space over F and a group that

p

will act as symmetries of spaces We refer back to IV which gave

V



H X X F U nS H

p

H BV

V

for a nite dimensional V CW complex X

We b egin by describing this in more conceptual algebraic terms In this section

we let U b e the category of unstable mo dules over the mo d p Steenro d algebra A

and let K b e the category of unstable Aalgebras Thus the mo d p cohomology

of any space is in K We shall abbreviate notation by setting H H BV

The celebrated functor T U U intro duced by Lannes is the left adjoint of

H for unstable Amo dules M and N

U M H N U T M N

ALGEBRAIC PRELIMINARIES LANNES FUNCTORS T AND FIX

Observe that the adjoint of the map M F M H M induced by the unit

p

of H gives a natural Amap TM M The key prop erties of the functor T

are as follows

The functor T is exact and commutes with susp ension

The functor T commutes with tensor pro ducts

This prop erty implies that if M is an unstable Aalgebra then so is TM The

resulting functor T K K is also left adjoint to H for unstable

Aalgebras M and N

K M H N K T M N

X is b oth an unstable Aalgebra and an H mo dule The Borel cohomology H

V

The action of H is given by a map of Amo dules and the bundle map

EV X BV

V

induces a map H H X of unstable Aalgebras We co dify these structures

V

in algebraic denitions Thus let H U b e the category of unstable Amo dules M

together with an H mo dule structure given by an Amap H M M For such

an H Amo dule M dene an unstable Amo dule FixM by

F H TM FixM F TM

p H TH p TH

The notation Fix anticipates a connection with Here we have used

to give that TH is an augmented Aalgebra and that TM is a TH mo dule TH

acts on H through the adjoint TH H of the copro duct H H H We

have another adjunction For unstable H Amo dules M and unstable Amo dules

N we have

H U M H N U FixM N

Comparing the adjunctions and we easily nd that for an unstable

Amo dule M

TM as unstable Amo dules FixH M

Less obviously one can also construct a natural isomorphism

H FixM as unstable H Amo dules H TM

TH

VI I I THE SULLIVAN CONJECTURE

The functor Fix has prop erties just like those of T

Fix H U U is exact and commutes with susp ension

The appropriate tensor pro duct in H U is M N

H

FixM FixN There is a natural isomorphism FixM N

H

Dene H nK to b e the category of unstable Aalgebras under H If M is an

unstable Aalgebra under H then its pro duct factors through M M and we

H

deduce from that FixM is an unstable Aalgebra If M is an unstable A

algebra then is an isomorphism of unstable Aalgebras If M is an unstable

Aalgebra under H then the isomorphism is one of unstable Aalgebras under

H We now reach the adjunction that we really want For an unstable Aalgebra

M under H and an unstable Aalgebra N

K FixM N H nK M H N

Lannes generalization of the Sullivan conjecture

Returning to top ology let X b e a V space Abbreviate

Fix X FixH X

V V

V

This is a cohomology theory on V spaces The inclusion i X X induces a

natural map

V V V

TH X H X X X Fix j Fix

V V

Here the middle isomorphism is implied by and the last map is an instance

of the natural map TM M The map j sp ecies a transformation of

cohomology theories on X By a check on V spaces of the form V W K one

nds that if X is a nite dimensional V CW complex then

V

j Fix X H X is an isomorphism

V

An alternative pro of using the lo calization theorem is p ossible In fact this must

b e the case the only way to reconcile and is to have an algebraic

isomorphism

F U nS M FixM

p H

LANNES GENERALIZATION OF THE SULLIVAN CONJECTURE

for reasonable M As a matter of algebra Dwyer and Wilkerson prove that there

is an isomorphism of H Aalgebras

H TM U nS M

TH

for any unstable H Aalgebra M that is nitely generated as an H mo dule Ten

soring over H with F this gives Combined with this gives an entirely

p

algebraic version of the isomorphism

V

U nS H X H BV H X

hV

V

of IV Here if M H X is nitely generated over H the isomorphism

V

agrees with that obtained by combining and Thus we may view

as another reformulation of Smith theory This reformulation is at the heart of

the Sullivan conjecture which is a corollary of the following theorem

Theorem Lannes Let X b e a V space whose cohomology is of nite

typ e and let Z b e a space with trivial V action whose cohomology is of nite

typ e Let EV Z X b e a V map Then the homomorphism of unstable

Aalgebras

Fix X H Z

V

induced by is an isomorphism if and only if the map

hV

Z X

p p

induced by is an equivalence

The map determines and is determined by a map

BV Z EV X X

V hV

of bundles over BV The map of the theorem is the adjoint via of

the map under H induced on cohomology by The map induces a map

EV Z X and the map of the theorem is its adjoint

p p

V V

To prove the Sullivan conjecture we take Z X and take EG X X

V hV

to b e the adjoint of the canonical map X X Then is the isomorphism

V hV

j of and X X is the map that Theorem claims to b e

p p

an equivalence Thus we see the Sullivan conjecture as a natural elab oration of

Smith theory

Theorem has other applications In the Sullivan conjecture we applied it to

obtain homotopical information from cohomological information but its converse

VI I I THE SULLIVAN CONJECTURE

hV hV

implication is also of interest Taking Z X and letting EV X X

b e the evaluation map the theorem sp ecializes to give the following result

Theorem Let X b e a V space such that the cohomologies of X and of

hV

X are of nite typ e Then the canonical map

hV

Fix X H X

V

is an isomorphism of unstable Aalgebras if and only if the canonical map

hV hV

X X

p p

is an equivalence

hV hV hV

When b oth X and X are pcomplete so that X X is the

p p

hV

X This is the starting identity we conclude that H X is calculable as Fix

V

p oint for remarkable work of Dwyer and Wilkerson in which they redevelop a great

deal of Lie group theory in a homotopical context of pcomplete nite lo op spaces

If we sp ecialize to spaces without actions and use we get the following

nonequivariant version of Theorem

Theorem Let Y and Z b e spaces with cohomology of nite typ e and let

BV Z Y b e a map Then the homomorphism of unstable Aalgebras

TH Y H Z induced by is an isomorphism if and only if the map

Z MapB V Y is an equivalence

p p

W G Dwyer and C W Wilkerson Smith theory and the functor T Comment Math Helv

W G Dwyer and C W Wilkerson Homotopy xed p oint metho ds for Lie groups and nite

lo op spaces Preprint

J Lannes Sur les espaces fonctionnels dont la source est le classiant dun pgroup e abelien

elementaire Publ Math I H E S

Sketch pro of of Lannes theorem

sketch the strategy of the pro of of Theorem The rst step is We briey

to reduce it to the nonequivariant version given in Theorem It is easy to see

that for a group G and Gspace Y we have an identication

hG

Y Map E G Y SecB G E G Y SecB G Y

G hG

G

where the right side is the space of sections of the bundle Y BG Let

hG

MapB G B G denote the comp onent of the identity map and MapB G Y

hG

SKETCH PROOF OF LANNES THEOREM

denote the space of maps whose pro jection to BG is homotopic to the identity

We have a bration

MapB G Y MapB G B G

hG

hG

with b er Y over the identity map

Now return to G V Here easy insp ections of homotopy groups show that

evaluation at a basep oint gives an equivalence

MapB V B V BV

and that the comp osition action of MapB V B V on MapB V Y induces an

hV

equivalence

hV

Y MapB V B V Map B V Y

hV

For a V space X the natural map EV X EV X induces a natural

p p

map X X and this map is an equivalence By the map of

p hV hV p

Theorem may b e viewed as a map

Z SecB V X

p p hV

The map determines a map EV Z X and this in turn determines and

p p

is determined by a map

BV Z X

p p hV

of bundles over BG The map is the comp osite map of b ers in the following

diagram of brations

MapB V Z SecB V X Z

p p hV p

BV Z MapB V B V Z MapB V X

p p p hV

MapB V B V MapB V B V

BV

The left map of brations is determined by a chosen homotopy inverse to

MapB V B V BV and the inclusion of Z in MapB V Z as the subspace

p p

of constant functions Clearly the middle comp osite is an equivalence if and only

if is an equivalence Applying Theorem with Z replaced by BV Z Y

VI I I THE SULLIVAN CONJECTURE

taken to b e X and replaced by the adjoint BV BV Z X of the

hV hV

comp osite map

BV Z MapB V B V Z MapB V X

hV

dened as in the middle row but b efore applying completions we nd that

the middle comp osite is an equivalence if and only if the induced map

TH X H H Z is an isomorphism Now gives an isomorphism

hV

H FixH X H TH X

hV TH hV

of unstable H Aalgebras Its explicit construction parallels the top ology in such

X H Z agrees with H This allows a way that the map Fix

H

V

us to deduce that is an isomorphism if and only if is an isomorphism

It remains to say something ab out the pro of of Theorem Since this is

nonequivariant top ology of the sort that requires us to join with those who use

the word space to mean simplicial set we shall say very little For a map

M N of unstable Aalgebras there are certain algebraic functors that one

st

may call Ext M N for xed t they are the left derived functors of a certain

K

t

functor of derivations Der N that is dened on the category of unstable

K

Aalgebras over N The relevance of the functor T comes from the fact that its

dening adjunction leads to natural isomorphisms

st st

Ext M H N Ext T M N

K K

for a map M H N of unstable Aalgebras with adjoint

There is an unstable Adams sp ectral sequence due originally to Bouseld and

Kan However the relevant version is a generalization due to Bouseld For a

map f X Y it starts from

p

st st

E Ext H Y H X f

K

and it converges in total degree t s to MapX Y f Under the hyp otheses

p

of Theorem the map Z MapB V Y induces a map of sp ectral

p p

sequences for any base p oint of Z that is given on the E level by the map

st st st

H Y H Ext TH Y F H Z F Ext Ext

p p

K K K

induced by TH Y H Z With due care of detail the deduction that

is an equivalence if is an isomorphism follows by a comparison of sp ectral

MAPS BETWEEN CLASSIFYING SPACES

sequences argument The converse implication is shown by a detailed inductive

analysis of the sp ectral sequence

An alternative pro cedure for pro cessing Lannes algebra to obtain the top ological

conclusion of Theorem has b een given by Morel Using a top ological interpre

tation of the functor T in terms of the continuous cohomology of propspaces

together with a comparison of Sullivans padic completion functor with that of

Bouseld and Kan he manages to circumvent use of the BouseldKan unstable

Adams sp ectral sequence and thus to avoid use of heavy simplicial machinery

A K Bouseld On the homology sp ectral sequence of a cosimplicial space Amer J Math

A K Bouseld Homotopy sp ectral sequences and obstructions Israel J Math

J Lannes Sur les espaces fonctionnels dont la source est le classiant dun pgroup e abelien

elementaire Publ Math I H E S

F Morel Quelques remarques sur la cohomologie mo dulo p continue des propespaces et les

V X Ann scient Ec Norm resultats de J Lannes concernant les espaces fonctionnels homB

Sup e serie

D Sullivan The genetics of homotopy theory and the Adams conjecture Annals of Math

Maps b etw een classifying spaces

We shall sketch the explanation given by Lannes in a talk at Chicago of how his

dsky that apply Theorem applies to give a version of results of Dwyer and Zabro

the Sullivan conjecture to the study of maps b etween classifying spaces Although

these authors apparently were not aware of the connection with equivariant bundle

theory what is at issue is precisely the map

a

G G G

B B MapE G B MapB G B

G

that we describ ed in VI I here the copro duct runs over RepG The

relevant theorem of Lannes is as follows

Theorem Lannes If G is an elementary Ab elian pgroup and is a

compact Lie group then the map

a

B MapB G B

p p

G

induced by is an equivalence

It should b e p ossible to deduce inductively that the result holds in this form

for any nite pgroup The original version of Dwyer and Zabro dsky is somewhat

VI I I THE SULLIVAN CONJECTURE

dierent and in some resp ects a little stronger although it seems p ossible to deduce

much of one from the other We say that a map f X Y is a mo d p

equivalence if it induces an isomorphism on mo d p homology We say that f is a

strong mo d p equivalence if it satises the following conditions

i f induces an isomorphism X Y

ii f induces an isomorphism X x Y f x for any x X

iii f induces an isomorphism

H X F H Y F

x p p

f x

for any x X where X and Y are the universal covers of the comp o

x

f x

nents of X and Y that contain x and f x

H H H

Say that a Gmap f X Y is a strong mo d p equivalence if f X Y

is a strong mo d p equivalence for each H G In view of VI I the following

statements are equivariant reinterpretations of nonequivariant results of Dwyer

and Zabro dsky and Notb ohm In nonequivariant terms when G their

G

results are statements ab out the map ab ove

and Zabrodsky If is a normal subgroup of a Theorem Dwyer

compact Lie group and G is a nite pgroup then the Gmap

B SecE G B is a strong mo d p equivalence

Actually Dwyer and Zabro dsky give the result in this generality for G Zp

and they give an inductive scheme to prove the general case when G

However their inductive scheme works just as well to handle the case of general

extensions Their result was generalized to ptoral groups by Notb ohm

Theorem Notbohm If is a normal subgroup of a compact Lie group

and G is a ptoral group then the Gmap B SecE G B

is a mo d p equivalence

However need not a strong mo d p equivalence in this case the comp onents

H

of induce injections but not surjections on the fundamental groups of corre

sp onding comp onents

These results are some of the starting p oints for b eautiful work of Jackowski

McClure and Oliver and others on maps b etween classifying spaces these authors

have given an excellent survey of the state of the art on this topic

MAPS BETWEEN CLASSIFYING SPACES

Lannes deduces Theorem from Theorem by taking Z B and Y

B The map is then the sum of the classifying maps of the homomorphisms

V sp ecied in VI I The deduction is based on the case X

of the following calculation

Theorem Let X b e a nite CW complex Then the natural map

Y

V

X TH X H

is an isomorphism where the pro duct runs over RepV

Proof The pro of is an adaptation of metho ds of Quillen Emb ed in U n for

some large n and let F b e the Gspace U nS where S is a maximal elementary

Ab elian subgroup of U n Quillen shows that the evident diagram of pro jections

X F F X F

X

induces an equalizer diagram

H X F H X F F H X

Let

j X TH X

and

Y

V

k X X H

Rep V

These are b oth cohomology theories in X Applied to our original diagram of

pro jections b oth give equalizers the rst b ecause the functor T is exact and the

second by an elab oration of Quillens argument We have an induced map from

the equalizer diagram for j to that for k The isotropy subgroups of the nite

CW complexes X F and X F F are elementary Ab elian and it therefore

suces to show that the map

Y

V

j W k W TH BW W H E

RepV

is an isomorphism when W is an elementary Ab elian subgroup of I learned

the details of how to see this from Nick Kuhn He has shown that T enjoys the

prop erty

Y

H BW BW TH

RepV W

VI I I THE SULLIVAN CONJECTURE

and the map in cohomology that we wish to show is an isomorphism is in fact

induced by a homeomorphism

a a

V

W E BW

RepV RepV W

To see the homeomorphism note that acts on the disjoint union over

V

HomV of the spaces W sends a p oint W xed by V to the

p oint W xed by the conjugate of It is not hard to check that as spaces

a a a

V V

W W W

HomV W V Hom RepV

V

Taking E as a mo del for each E this implies the required homeomor

phism

W G Dwyer and A Zabro dsky Maps b etween classifying spaces Springer Lecture Notes in

Mathematics Vol

S Jackowski J McClure and B Oliver Homotopy theory of classifying spaces of Lie groups

In Algebraic top ology and its applications ed Carlsson et al MSRI Publications Vol

Springer Verlag pp

N J Kuhn Generic representation theory and Lannes T functor London Math Scc Lecture

Note Series Vol

J Lannes Cohomology of groups and function spaces Notes from a talk at the University of

Chicago Preprint Ecole Polytechnique

D Notb ohm Maps b etween classifying spaces Math Zeitschrift

D G Quillen The sp ectrum of an equivariant cohomology ring I I I Annals of Math and

CHAPTER IX

An intro duction to equivariant stable homotopy

V M

G

Gspheres in homotopy theory

n

What is a Gsphere In our work so far we have only used spheres S which

have trivial action by G Clearly this is contrary to the equivariant spirit of our

work The full richness of equivariant homotopy and homology theory comes from

the interplay of homotopy theory and representation theory that arises from the

consideration of spheres with nontrivial actions by G In principle it might seem

reasonable to allow arbitrary Gactions However a closer insp ection of the role

of spheres in nonequivariant top ology b oth in manifold theory and in homotopy

theory gives the intuition that we should restrict to the linear spheres that arise

from representations Throughout the rest of the b o ok we shall generally use the

term representation of G or sometimes Gmo dule to mean a nite dimensional

real inner pro duct space with a given smo oth action of G through linear isometries

We may think of V as a homomorphism of Lie groups G O V This

convention contradicts standard usage in which representations are dened to b e

isomorphism classes

For a representation V we have the unit sphere S V the unit disk D V

V

and the onep oint compactication S G acts trivially on the p oint at innity

V V

which is taken as the basep oint of S The based Gspheres S will b e central to

virtually everything that we do from now on We agree to think of n as standing

n n

for R with trivial Gaction so that S is a sp ecial case of our denition For a

based Gspace X we write

V V V V

X X S and X F S X

IX AN INTRODUCTION TO EQUIVARIANT STABLE HOMOTOPY

V V

Of course is left adjoint to

When do we use trivial spheres and when do we use representation spheres

This is a subtle question and in some of our work the answer may well seem coun

terintuitive In dening weak equivalences of Gspaces we only used homotopy

groups dened in terms of trivial spheres and that is unquestionably the right

choice in view of the Whitehead theorem for GCW complexes Nevertheless

there are homotopy groups dened in terms of representation spheres and they

often play an imp ortant role although more often implicit than explicit We may

think of a Grepresentation V as an H representation for any H G For a based

Gspace X we dene

V H V

X S X G S X

H H G

V

Here the brackets denote based homotopy classes of based maps with the ap

propriate equivariance For a pair X A of based Gspaces we form the usual

homotopy b er F i of the inclusion i A X and we dene

H H

F i X A

V V

It is natural to separate out the trivial and nontrivial parts of representations

Thus we let V H denote the orthogonal complement in V of the xed p oint space

H

V We then have the long exact sequence

H H H H

X X A A X

V H n V H n V H V H

H

X and of p ointed sets thereafter of groups up to

V H

Waner will develop a GCW theory adapted to a given representation V in the

next chapter and Lewis will use it to study the Freudenthal susp ension theorem

for these homotopy groups in the chapter that follows There is a more elementary

standard form of the Freudenthal susp ension theorem due rst to Hauschild that

suces for many purp oses Just as nonequivariantly it is proven by studying

V V

the adjoint map Y Y Here one pro ceeds by reduction to the non

equivariant case and use of obstruction theory Recall the notion of a equivalence

from Ix where is a function from conjugacy classes of subgroups of G to the

integers greater than or equal to Dene the connectivity function c Y of

H H

a Gspace Y by letting c Y b e the connectivity of Y for H G we set

H H

c Y if Y is not path connected

GUNIVERSES AND STABLE GMAPS

V V

Theorem Freudenthal suspension The map Y Y is a

equivalence if satises the following two conditions

H H

H c Y for all subgroups H with V and

K K H

H c Y for all pairs of subgroups K H with V V

Therefore the susp ension map

V V V

X Y X Y

G G

H H

is surjective if dimX H for all H and bijective if dimX H

H Hauschild Aquivariante Homotopie I Arch Math

U Namb o o diri Equivariant vector elds on spheres Trans Amer Math So c

GUniverses and stable Gmaps

We next explain how to stabilize homotopy groups and more generally sets of

homotopy classes of maps b etween Gspaces There are several ways to make this

precise The most convenient is that based on the use of universes

Definition A Guniverse U is a countable direct sum of representations

such that U contains a trivial representation and contains each of its subrepresen

tations innitely often Thus U can b e written as a direct sum of subspaces V

i

where fV g runs over a set of distinct irreducible representations of G We say

i

that a universe U is complete if up to isomorphism it contains every irreducible

representation of G If G is nite one example is V where V is the regular

representation of G We say that a universe is trivial if it contains only the trivial

G

irreducible representation One example is U for a complete universe U A nite

dimensional sub Gspace of a universe U is said to b e an indexing space in U

We should emphasize right away that as so on as we start talking seriously ab out

stable ob jects or sp ectra the notion of a universe will b ecome imp ortant even

in the nonequivariant case

We can now give a rst denition of the set fX Y g of stable maps b etween

G

based Gspaces X and Y

Definition Let U b e a complete Guniverse For a nite based GCW

complex X and any based Gspace Y dene

V V

fX Y g colim X Y

G V G

IX AN INTRODUCTION TO EQUIVARIANT STABLE HOMOTOPY

where V runs through the indexing spaces in U and the colimit is taken over the

functions

V V W W

X Y X Y V W

G G

V V

that are obtained by sending a map X Y to its smash pro duct with the

W V

identity map of S

When G is nite and X is nite dimensional the Freudenthal susp ension the

orem implies that if we susp end by a suciently large representation then all

subsequent susp ensions will b e isomorphisms

Corollary If G is nite and X is nite dimensional there is a represen

tation V V X such that for any representation V

V V V V V V V

0 0 0 0

X Y Y X

G G

is an isomorphism

Let X and Y b e nite GCW complexes If G is nite the stable value

V V

0 0

Y fX Y g X

G G

is a nitely generated ab elian group However if G is a compact Lie group and

X has innite isotropy groups there is usually no representation V for which

V

all further susp ensions are isomorphisms and fX Y g is usually not nitely

G

generated

V

Remark The groups fS X g are called equivariant stable homotopy

G

G

groups of X and are sometimes denoted X However it is more usual to

V

G

denote them by X relying on context to resolve the ambiguity b etween sta

V

ble and unstable homotopy groups

The denition of fX Y g just given is not the right denition for an innite

G

complex X Observe that

V V V V

X Y X Y

G G

Definition Let U b e a complete Guniverse For a based Gspace X

dene

V V

QX colim X

V

where V runs over the indexing spaces in U and the colimit is taken over the

maps

V V W W

X X V W

EULER CHARACTERISTIC AND TRANSFER GMAPS

V V

that are obtained by sending a map S X S to its smash pro duct with the

W V

identity map of S Observe that the maps of the colimit system are inclusions

Lemma Fix an indexing space V in U For based Gspaces X there is a

natural homeomorphism

V V

Q X QX

W W

Proof Clearly QX is homeomorphic to colim X and similarly for

W V

V V

Q X By the compactness of S and the evident isomorphisms of functors

W W V W V V W V

for V W and

V W V W V V V W V W V V W W

colim X colim X colim X

V The conclusion follows where the colimits are taken over W

Lemma If X is a nite GCW complex then

fX Y g X QY

G G

Proof This is immediate from the compactness of X which ensures that

V V

X QY colim X Y

G V G

For innite complexes X it is X QY that gives the right notion of the stable

G

maps from X to Y We shall return to this p oint in Chapter XI I where we

intro duce the stable homotopy category of sp ectra

Euler characteristic and transfer Gmaps

We here intro duce some fundamentally imp ortant examples of stable maps that

require the use of representations for their denitions The Euler characteristic

and transfer maps dened here will app ear at increasing levels of sophistication

and generality as we go on

Let M b e a smo oth closed Gmanifold We may emb ed M in a representation

V say with normal bundle We may then emb ed a copy of as a tubular neigh

b orho o d of M in V Just as for nonequivariant bundles the Thom complex T of

a Gvector bundle is constructed by forming the b erwise onep oint compact

ication of the bundle letting G act trivially on the p oints at innity and then

identifying all of the p oints at innity to a single Gxed basep oint We then

have the PontrjaginThom map

V

t S T

IX AN INTRODUCTION TO EQUIVARIANT STABLE HOMOTOPY

It is the based Gmap obtained by mapping the tubular neighb orho o d isomor

phically onto and mapping all p oints not in the tubular neighb orho o d to the

basep oint The inclusion of in where is the tangent bundle of M

M M

induces a based Gmap

V

M S e T T

M

The comp osite of these two maps is the transfer map

V V

M e t S M

asso ciated to the pro jection M fptg which we think of as a trivial Gbundle

Of course this pro jection induces a map

V V V

S M S

We dene the Euler characteristic of M to b e the based Gmap

V V

M M S S

The name comes from the fact that if we ignore the action of G and regard M

as a nonequivariant map of spheres then its degree is just the classical Euler

characteristic of M The pro of is an interesting exercise in classical algebraic

top ology but the fact will b ecome clear from our later more conceptual description

of these maps In fact from the p oint of view that we will explain in XVx this

map is the Euler characteristic of M by denition

Since V is not welldened we just chose some V large enough that we could

emb ed M in it it is most natural to regard the transfer and Euler characteristics

as stable maps

M fS M g and M fS S g

G G

Observe that when M GH the map GH of can b e written as the

comp osite

e t

V V V

W

GH S G S G S GH S

H H

where W is the complement of the image in V of the tangent plane LH at

the identity coset and e is the extension to a Gmap of the H map obtained by

L H W

smashing the inclusion S S with S The unlab elled isomorphism is

given by I

MACKEY FUNCTORS AND COMACKEY FUNCTORS

More generally for subgroups K H of G there is a stable transfer Gmap

GK GH asso ciated to the pro jection GH GK In fact we

may view as the extension to a Gmap

G K H GK

K

of the pro jection K H fptg and we may construct the transfer K map

K H starting from an emb edding of K H in a Grepresentation V regarded as

a K representation by restriction We then dene to b e the map

V V V V

GH S G S G K H S GK S

K K

where the isomorphisms are given by I and the arrow is the extension of the

K map K H to a Gmap Note that any Gmap f GK GH is

the comp osite of a conjugation isomorphism c GK Gg K g and the

g

pro jection induced by an inclusion g K g H We let c c With these

g

g

denitions we obtain a contravariantly functorial assignment of stable transfer

maps f to Gmaps f b etween orbits Of course such Gmaps may themselves

b e regarded as stable Gmaps b etween orbits

Mackey functors and coMackey functors

Ggraded homology and cohomology We are headed towards the notions of RO

theories but we start by describing what the co ecients of such theories will lo ok

like in the case of ordinary RO Ggraded theories

Recall that the ordinary homology and the ordinary cohomology of Gspaces

are dened in terms of covariant and contravariant co ecient systems which are

functors from the homotopy category hG of orbits to the category A b of Ab elian

groups Let A denote the category that is obtained from hG by applying the free

G

Ab elian group functor to morphisms Thus A GH GK is the free Ab elian

G

group generated by hG GH GK Then co ecient systems are the same as

additive functors A A b

G

Now imagine what the stable analog might b e It is clear that the sets fX Y g

G

are already Ab elian groups

Definition Dene the Burnside category B to have ob jects the orbit

G

spaces GH and to have morphisms

B GH GK fGH GK g

G G

IX AN INTRODUCTION TO EQUIVARIANT STABLE HOMOTOPY

with the evident comp osition We shall also refer to B as the stable orbit cate

G

gory Observe that it is an A bcategory its Hom sets are Ab elian groups and

comp osition is bilinear

We must explain the name Burnside The zeroth equivariant stable homotopy

group of spheres or equivariant zero stem fS S g is a ring under comp osition

G

We shall denote this ring by B for the moment It is a fundamental insight of

G

Segal that if G is nite then B is isomorphic to the Burnside ring AG Here

G

AG is dened to b e the Grothendieck ring of isomorphism classes of nite Gsets

with addition and multiplication given by disjoint union and Cartesian pro duct

For a compact Lie group G tom Dieck generalized this description of B by

G

dening the appropriate generalization of the Burnside ring In this case AG

is dened to b e the ring of equivalence classes of smo oth closed Gmanifolds

where two such manifolds are said to b e equivalent if they have the same Euler

characteristic in B again addition and multiplication are given by disjoint union

G

and Cartesian pro duct An exp osition will b e given in XVI Ix

Definition A covariant or contravariant stable co ecient system is a co

variant or contravariant additive functor B A b A contravariant stable

G

functor A covariant stable co ecient sys co ecient system is called a Mackey

tem is called a coMackey functor

When G is nite Dress rst intro duced Mackey functors using an entirely

dierent but equivalent denition to study induction theorems in representation

theory We shall explain the equivalence of denitions in XIXx The classical

examples of Mackey functors are the representation ring and Burnside ring Mackey

functors which send GH to RH or AH The generalization to compact Lie

groups was rst dened and exploited by Lewis McClure and myself

Observe that we obtain an additive functor A B by sending the ho

G G

motopy class of a Gmap f GH GK to the corresp onding stable map

Therefore a covariant or contravariant stable co ecient system has an underly

ing ordinary co ecient system Said another way stable co ecient systems can b e

viewed as given by additional structure on underlying ordinary co ecient systems

What is the additional structure Viewed as a stable map GH is a mor

phism GG GH in the category B and more generally so is f for any

G

Gmap f GH GK We shall see in XIXx that every morphism of the

category B is a comp osite of stable Gmaps of the form f or f That is the

G

extra structure is given by transfer maps When G is nite we shall explain alge

MACKEY FUNCTORS AND COMACKEY FUNCTORS

braically how comp osites of such maps are computed In the general compact Lie

case such comp osites are quite hard to describ e For this reason it is also quite

hard to construct Mackey functors algebraically However we have the following

concrete example It may not seem particularly interesting at rst sight but we

shall shortly use it to prove an imp ortant result called the Conner conjecture

Proposition Let G b e any compact Lie group There is a unique Mackey

B A b such that the underlying co ecient system of Z is con functor Z

G

stant at Z and the homomorphism Z Z induced by the stable transfer map

GK GH asso ciated to an inclusion H K is multiplication by the Euler

characteristic K H

Proof In XIXx we shall give a complete additive calculation of the mor

phisms of B from which the uniqueness will b e clear The problem is to show

G

that the given sp ecications are compatible with comp osition We do this indi

Thought of rectly As already noted we have the Burnside Mackey functor A

top ologically its value on GH is

fGH S g fS S g B

G H H

and the contravariant functoriality is clear from this description Dene another

Mackey functor I by letting I GH b e the augmentation ideal of AH Thought

of top ologically its value on GH is the kernel of the map

Z fGH S g fG S g

G G

induced by the Gmap G GH that sends the identity element e to the

coset eH Using XIX and the denition of Burnside rings in terms of Euler

is a subfunctor of A A key p oint is the characteristics one can check that I

identity

Y H K H Y

K

of nonequivariant Euler classes for H K and H spaces Y One can then dene

to b e the quotient Mackey functor AI the desired Euler characteristic formula Z

can b e deduced from the formula just cited

T tom Dieck Transformation groups and representation theory Springer Lecture Notes in

Mathematics Vol

A Dress Contributions to the theory of induced representations Springer Lecture Notes in

Mathematics Vol

L G Lewis Jr J P May and M Steinb erger with contributions by J E McClure Equiv

ariant stable homotopy theory Vx Springer Lecture Notes in Mathematics Vol