Equivariant Algebraic Topology and the Equlvar lant Brown Reprcscntabili ty Theorem

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St.John's

EquivariantAlgebraic Topology andthe Equlvarlant Brown Reprcscntabili tyTheorem

by

©PhilipHoskins

A thesis submitted tothe SchoolofGraduate Studiesin partial fulfillmentof the

requirements for thedegree of Maslc,('fScience

DepartmentofMathematics and Statistics Memorial UniversityofNewfoundland

1912

Newfound land

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Abstract

Themain purpose of this thesis is to give a COllJlltctcproof ofthe Eqnlvariant Drown RcprcscntabiliryTheorem.in the processdevelopingth~equh'arianl algebr,l ic topology neededin the final proof. The proofof the th c-oretu in the category of path-connected G-spaces isgiven in Chapter·1and followsthe proofoflhe non- cquivaelcntcasegivenin Spanier ([Sp], pp-,jOG-411).Thereis another account of the proal given in Switzer ([Swl.pp.152-157) ,which iscloser tothe originalaccount given by Brown[DrI ], Equlvarlant versions ofthetheorem are announcedin [LMS]

and [V], forexample,butno details of the proofsarc given.

InChapter1, the basic theoryof G-spaces and G-mapsispresented.G-finalaut!

G-initialstructures ona sel are defined and sufficient conditionsaregiven which allowsuch G-spacestructuresto be constructed.

The equi\'arianthomotopygroups are defined in Chapter:1and the isomorphisms 1r~f(){)elfn{X1t )and1r~/(X,A ):!!l:"n{XII,All)are established .Thesetw<)results arc then usedto provethe resultsabout equivaelnnt homotopygroupsneededin Chapter 4.

In Chapter 3, G·CW-complexesare defined and ailthenecessney homotopic prep- erticsof G-CW-complexesarc developed,culminatingin the proof ofthe E'lllimri,lllt Whitehead Theorem,

Finally, in Chapter 5, we prove an equivariant versionof the stetcmcntlha Lif,1 Junctor satisfies thewedgeAxiom andthe Mayee-Vletoels Axiom givenillBrown's originalversion ofhistheorem, then italso satis flcsthe EqualizerAxiom givenill Spanier's version or the theorem.This immediately gives a Switecr styleverslon of thcmainrcsult.

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Acknowledge ments

Iwould liketothank theSchool or Gradua teStudiesand the DepartmentofMath- ematic and Statistics fortheirfinanci~supportdurinr;thewritinr; of this thesis.I would especially liketotha nkDr. PeterBooth for his suucstions,encouragement, and valuableadvice givenduring ournumerousdiscussionsconcerninsthe topicsand rcsuhs presented in the thesis.

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iii Tableor Cont en ts

Abstract ... ... .. ...••.• •.• •• .•.•..•. .. ...• ... ..i Acknowledgernt'nh... ...•.•.••.• ••...•.. .. .. ..•. ..•.•. .•ii Chapter1 -Equivariant InitialAnd FinalG·,pac~

Stru ctures.... ... ..•. •.•... •... ...•.•...•....••.•.. •.... .I

Chapter2 -EquivariantHomotopyGroups 21

Chapter3•G·CW· ComplexcsandtheEquivaria nt

\Vhitehea.d.Theorem . 29

Chapter,I•Equivariant Drown Rcprcsentu billty

Theorem•. . . ... .•. . .. . .. . . ... . . . .. . .• ... . . .. . ....•. ·1,\

Chapter 5·The~h,)'Cr.VictorisConditio nand

the EqualizerAxiom 56

References..• . .•..• ....•...•. . .... . . .. ..• .... •. •. ... . ... GO

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Chapter1 Equivariant Initial and Final G-spa.-cc Structurcs Let G be atcpologlcal group and let X beatopologicalspace. The nXis called 11.G"~Jlaceifthereis amap(continuous functio n)

9:GxX _X

suchthat thefollowing conditionsarcsatisfied:

(i)9(9lt9(91,r))=9(9.91,X)for91,92fG, xlX (ii)9(c,r)=xfor allXi X ,whcrceis the identit yinG.

Usually9(9,X)willbedenotedby9r,sc that (i) and (ii)become:

(i)9d92X)

=

(9192)rfor91,92fG,:leX (ii)ex= xfor all:lc'(

Sneha~is calledanadionofGonX.

Foreach

«x,

Gx={gX!gf G}is called the2I:liliof:l.If

s ;

Xthen

as

=~Gr.

AsubsetA~Xisfi.mlby GifGA=A.

IfX and }' are G-spacesthe namap

f :

X- }'is saidto be eqliivariantor a~if for~11:lc'(,geOwe have1(9x)

=

9f(x).Hencefis a G-mar;f.'~ following diagram commutes

GXX..!!... X

1 Q X 11 11

Gx l ' _ Y

.,

where~xand~I'MetheactionsofGonXand}' respec tively, 1.1Proposition. The composit ion oftwoG-mapsisaC·ma p.

.r..mcl:Assume

f :

X _Yandh:l'...ZarcG-maps. Then

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(h o/)(g z ) h(J (gz )) h(g/(z)) gh(J(z)) g(h oll(z) •

Examples of a·spaces 1. Trivial a·spaces

Any topologicalspaceXcanbe given the trivialncricnofG,defined hy

¢:GxX _ X

~(g,:r:)

=

e,

In particula rthespacesE"

=

{uR"1[e] ::;I}, S"

=

{x£Rn+l l

Ix l

=I}. and In={(XhX2'''',x..)tR"IO:c::Xi:$1, 1:$i::;n]willalways be giventhe trivial actionoie.

2.Coset or homogeneousG-spaces

IfHis a closedsubgroupofGandGislocally compact Ile uadorff thentile~ctof left cosets~.withthe quotienttopology,is a G-space withactiondefinedby

¢:Gx j

¢(g',g H ) G

Ii

g'gll To shew that

4t

is contin uousconsider the composite

GXG~Gx * .-t.. ]7

whereqisthe quotientmap. SinceG is locallycompactHausdorffandq :G ...~ isan identification,itfollows thattexqisalso an identification($CCfor example

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[nro),4.:1.2,p.101). So,p iscontinuous if anJonlyif~o(10xq)is continuous.But

~0(I GXq)is continuoussince it is equalto the composite

GXG~G-!...*

wherem(g',g)=g'gisthe group multiplicati on.

Fortheremainder ofthe thesisweassumeGis compact Hausdorff .Itisknown (srefor example(0],Proposition3.3,p.23)that~isHausdorff if and onlyifHis c[osl.-il inG.Soilwill alsobe convenient to henceforthassumeallsubgroupsHofG referredtoare closed.

3. Funclion Space!

LetMapo(X, Y)denot ethespace of a-maps from theG-spaceXtothe a·space Y.wilhthecompact opentopology.Thisspace canbegivenaC·actiondefinedby

GxMapo(}<,Y) (9,f)

Mapo(X,Y ) (r- g/(g- Ir

»)

H }"islocallycompact Hausdorff,then we bave the followingexponential law for C·spaccsand C-maps. There is a bijection

definedby

i-« t

wherej(%)(y)

=

f(%,y)[see[D),^p,35).

G· inili~ 15 l[llet !! rrs

IfX is aset, then a G-space st ruct ure onXis atopology onXalong with a continuousactionorGonX.

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LetJbe an indexin gsetand let

{Ii :

X - Xj}j'Jhea familyof functions fromthe setX to thetopological spaces

P'..

}j.J.Thena topologyon X issaid to belnilialwith respectto{ljli e)ifitsatieflcsthefullowinguniversal prcpcrtj..;

foranytopological spaceZ,a funct ionh: Z_Xiscontinuousifandonlyifeach compositeliohiscontinuou s. Suchittopologyexist s;it is the topology with eubbasis consisti ngofall setsof the form{fr¹(U)IUopeninX,jcJ}.{For morc detnllssec [Brc],p.153).

Now let{Ii:X - Xj} j<J,as above,beafamily of G-maps.Theil the G-SPil.t·c str uct ure onXis sa id to bc.G:..ini.1lalwithrespectlath efunction s{Jilj,)Hitsa lisfk':l lhcfollo winguniverse! property : for anyG-s paceZ,a functionp:Z...X isaa-map ifandonlyifeach compos ite

Ii

0pisa G-m:tp.

Wealso note tha tirp : Z->XisaG-m.lp itfollowsfromProposition1.1 that thecomposi te

h op

is a G-map for eachj(J.Hence,tocstn hlishthatXhn.~the G-initialstructure wit h respecttothefamilyofG-maps{I,:X->

X ,Ld.

^weonly needto show thatthe conti nuityof p followsfromthe continuit yoftheeomposh lons

hop,

for alljtJ . Example:ProductofG-spaces

For anyIarnl!v of G·spa ces{Xi}JtJtheprodu ct

II

X,ca llbe given a (;·initi.ll

,.J

structurewithrespectto the projectionsPj:

II

X,... Xj,wherenjdX,hasthe jrJ

initial topology.The act ion ofGon

II

Xjisthe diagOlI::LIactiondelinedbyrP:Gx jd

(Il X ') ... PX

jwhere

~(g, (zjlicJ))

⁼(gz, lj , J ).Toshow thal¢

i~

continuous,

j.J J.J

considerthefollowingdiagram

Gxni,JXi

---.!....

nj.JXj

'o." j j"

"

^Xi

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wherefjhithe action ofGon Xj. ThcnearncompositePj0

if!

is equalto the eomposite

?j

0(Ia^X^Pj)which is clearly continous.By the universal propertyCor initialtopologies,; is continuous.We noticethat the projectionsPj:

IT

Xj-+ Xj

jeJ are G-maps.

Leth:Z-+

II

Xjbe a functionfrom the G-spaceZto

11

Xjsuch thatPj0his

~ ~J

a G-maplor eachicl,Thcnfor allZlZ, glG wehave

gh(z) (gp;(h(z))U<J) (g(p;0h)(')U<J)

«p;0h)(g, )lj <J) (p;(h(gz»Jj<J) h(g,)

Thus h is a G-mapandthe G-space structureon

II

Xj is G-initialwithrespect to

jd

b'j)jeJ'

Sufficientconditions for the generalcase are given in the followingresult.

1.2 Proposition.Let{h :X-+Xj}jeJ bealamily clIuncticns Irom thesetXto the G-spaces{X j} joJ.ThenXcanhegivena G-initialstructu re withrespect to{h}jcJ ifthe following conditionsare satisfied:

(i) the functionh:X-+

IT

Xjdefinedbyh(z}=(fj(z )lil1 ),isinjective, and j.J

(ii)h(X) is fixed by G,i.e.iCglG andylh(X),then gylh(X).

P.mcl:

LetXbe giventhe initialtopology with respectto(h }joJ.We have to define

allaction of G onXsuch that the universalpropertyholds.

Using condition(ii)we see that, foreach9lG, :uX, there existsanilX with 1I(i) =gh(z),lind thisi is unique by condition(i).Define

if!:

GxX-+Xby

~(g,.r)

=

^i,\Ve have to show that~is continuous.

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itfollow! bythetransitive rule forinitialtop ologies (see (Ora l,5.6.8,p.15-1)tha t Xhastheinitial topologyrcla li'..c to the inj ecti onh,and soh: X_ .\ (X )isa homeomorp hism.

Let0bethe diagonalaction011h(X).The n¢

=

h-^I0(IGXh)0h,i.e.weha ve thefc llcwingecmmutativediagram,

a

xh(X)

~

h(X)

' '''I j, -,

GxX --;- X

So4>is continuo ussinceitis the composite of continuousfunc tio ns.

Letp:Z_ Xbe a functionfromthe G·spaceZtoX.Assume hopisaG-map Coreachjf}.Let~x,¢zbe the actions of0onXandZ,respe ctively.ThenI'is11.

G· ma p irpoQ>z

=

9xo(lGxp).Outfro m theargumentabovetPx

=

h- 1o Oo(IGXh).

So we have to show thallheouterperi meterof thefollowingdlagrnmcom mutes:

GxZ

~

Z

GxX X

"

'·"1 f ,-'

a

x"( Xl

~

^h(X)

Let(9,Z)' 0xZ.Then

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0(10Xh)(IGXp)(g,,) O(IGxh)(g, *ll O(g,(!i(p(,»)lj, Jll (g!i(P{,lllj<J) (glli op)(')/j<J) (Ui0p)(g' l lj<J) (!;(p(g'll!i <J) h(p(g, )) hp9z(9,z) SoJl(gz )=gp(z)andpis a G-map .• Examples ofa·initialstructuresandProposit ion1.2

1.ProductG-spAces.

Theproducttopology isaspecial caseof theabove proposition withhthe identi ty. Specialexamplesof productG·spaccsarc

~

^x^e,

*

^x^Sri,

^~

^x^En,

and

%

^xI, whe reG acts triviallycn ",S",En,and1.

2. G-subs paccs.

IfAisa subse tofXthatisfixedbyG, thentheinclusioni :A-+Xsatisfies the conditionsofProposition1.'2whereh "" i.Soif

? :

GxX _ Xisthe action ofGonXthen ¢IG xA--IAistheaction ofGonAwhichgivesA theC·jnit i,,1struct ure011A,whereAhas the subspacetopology.Aisthen calleda G, sllh spacc ofX.

If(X,. )is abasedspace withbasepoint

*,

then(X,.)willhecalled a baseda-spa ce iftheactionofG onXis suchthat*isfixedbyG.

ForG-subspa ceswehavethefollowing:

1.3Propositi on.Let

f :

X - }'bea G-map .Then

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(i)HAis a C-subs pace ofX,then/(A )isa.C·subsp;\ce of}' , (ii)IfBisaC-subspaceof}',then /-1(8)isaC·subspa.c:e ofX.

fulgf,

(i)Wehavetoshow tha.tGf(tl)

=

/(A ).LetileA,gtG.Theng/(lI)

=

/(go)e/(,\) since ga(A.HenceG/(/l)~/(A)andsincee/(,i)=/(A)wehaveG/(A)

=

I(A).

(ii)Let:uj- t( 8)andgeG.Then/(g%)==g/(:)(8since/(%)e8andGil==/J.

Ilenceg:uj-t(B) ,Cr l(D)c;;;:/- t(B) andG/-1(D)

=

/- I( D).• C_fio;\!strl1 ct ll rr~

LetJbe anindexing set andlet

{Ii :

Xj--0XL,JbeII.Familyoffunctions Irom the spaces{Xj}itJtothesd X.Thena topologyonXiss(lidtobe

Iinnl

wit h respect to thefunct ions{fj).i.Jifthe followinguniversa]propertyissatis fied; for anytopologicalspace Z,afunct ionh:X_ Ziscontinuousif andonlyif('a(1I compositeh0

h

is continuous. Thefinaltopology existsand coosistsofa.1Isets U~Xsuch Lhat/Tt(U)is openinXjfer alljcJ.

Nowlet {Ii : Xj_ X)itJoasabove,beafamilyofG·m.t ps.TheilaC-sl',' ce st ructure onXis said to!L:.fin4.1withrespectto{fj}j.Jifitsatisl'iesthefollowing universalprope r ty:for any C·spaceZ,afunction11:X....,ZisilG-map ifand onlyifeachcompositeh0

Ii

is&C-map.AgainwenotethatfromProposition1.1it followsthat ifh:X_ZisaC-map ,thenthecomposite 110

h

isalsoaG·ma pfor eachjcJ.Hen ce, toes tablishthatXhastheC-finalstructure withrespecttothe familyof G-m aps{fj:Xi- XI;.J,weonlyneedtoshowthatthe continuityofh followsfrom thecontinuity of thecompositeshoI;.for alljeJ.

Example:Sum of C·spa ces

For anyfamily of C·space{Xj }itJthe topologica lsum

UX

i•i.e.the disjoinl itJ

union of the spacesXjwiththe usual final topology,canhe givena G-final struct ure with rcspecttotheinclusionsii:Xj...

U

Xj.The actionof G on

U x,

isdefine.J

jtJ j.J

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by:

. ,ax U x;

i<J (g,ri)

u ^x;

j<J

gXi forg(G,rj!Xj, and[c-l To show that

,p

is continuou swe note that sinceGis compactHausdorff,Gx

U

Xi

i,J has thefinaltopologywit hrespect toIc xii:GxX j-+Gx

U

Xj(see[Bre],4.3.2,

j,J

p.101).Then

f

is continuous since each composite

e

c(Iaxij)=ii~j.whe re¢jis the actionofGonXi'We noticethatthe inclusionsii: Xi....

U

XiareG-maps

jr:,J lor alljd.

Nedleth :

U

Xj...Zbe a functionfrom

U

Xjto the G·spaceZwithh0ija

i,J j<J

C-lIIapforall[cl :ThenwehaveforuXh gcG:

h(gr ) hii(gz) (hoi;)(gx) g(hoi;)(x) gh(i;(x)) gh(x) Thustile universalpropertyis satisfied.

Sufficientconditionsfor theexistence orG-fioalstructuresaregivenin the following result.

1AProposit ion .Let{Ii:Xj-+X}i<Jbe afa mily offunct ions fromthe G·spaccs {Xj} j<Jtothe setX.ThenXcan be given aG-linalst ruclurewithrespect to{fj}i<J if thefollowing conditionsarc satisfied:

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(i)thefunctionp:

UX

^j^{- .}^X^definedby p(z;)==f;( z ,)forr,l.\;,is su rjec tive ,

jf J

aod

(ii )Ifp(z;)==p(r j)forXiC\'";.%j (.1( j,withi,it J,thenp(gzi)== 11(9Xj)forallgtG.

P..mszf;From(i)itfollows thitt for each,1:(.\' ,the reisXCUXjsuch1hillp(i)==x,

,it}

Definethe actio n ofGon Xby:

s-c cx-: »

.(9•• )=p(g. )

This action iswell-de finedbycondit ion(ii)

I£X is giventhefinaltopology wit h respectto thefunctions{!i}jrJ.then I':

U

Xj-+Xisan identi ficat ion. SinceGis locally compactHausdorlT j<J

lax p:G xUXj -t Gx X

jt}

is alsoan identification.Henceit followsfromthefollowing commut a tive dlngrnm.

GxUj </ Xj

~

^G^{x X}

.j l -

x

tha t~iscontinuous.

Leth:X ...Zbea funct ionfromXto theG·spaceZ.Assumeh0

Ii

isaa· map (Ofeachill.Let:rcX.Thenwe havetoshowh(gr)==gh(z)for any9,0.

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II

h(gz ) h(p(gi ))wherei()(jIorsomejfJ h(f;(g' ))

(h o !;)(y.)

g(ho/j)(i) sinceeach h0

Ij

isaG-map gh(f;('»)

gh( x) Sohis'"C-map.•

Examples ofC-fina lstructuresandProp ositio n 1,4

I.Topologicalsums Atopological sum

U

Xjofa·spacesis coveredbythe j<J

abo ve proposit ionwit hptileid entity on

U

Xj' jcJ 2.Quo tientG-space

Lcte-bean equ ivale nce rel ati ononthea-spaceXsuchthatifoX- 1Ithen gr ....911forallx,y£X,9£0.The nby Prop o sit io n1.4,the quotie nt spa ce~ ca nbe give n theG·finalst r uctu rewiththe actionon~definedby

9:GX~ ""~

'(g,!.])

=(g.]

Thepof Propositi on1.5 isthe obvio us ide nt ifica ti on mapX-+~.

In pa rtic u la rifAis a G-s ubspace ofXthen :},the quoti entspace obtained byshrink ingAtoa.point ,CAnbe give nthe C· final struct u rewit hrespe ctto the quotient mapp:

X-f.

Given a based set(Z,.),therule

f .-. Jp.

whe re

f

is abased fu nc tio n }~/A-+Z.determinesabiject ive co rresp on d en ce betweenbasedfun el ions }~/A_ ZandIuuct ioneY-+ZwhichtakeAto

*.

He nceit followsfrom Pro positio n 1..1 th a tpsatisfiesthefollowinguniversal pro perty:

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12

IfZi,abued a·space,thcnlberulef_ lP,where! :(YI A,{A }) ...

(Y,')is "basedG·map.dct<'l'minesabijcetive eorrespcndence between

(i)b...,j G· ma.. ll'/ A. {All-(Z,.),",d (ii)G·map5ofpairs(Y,II)...(Z.-).

If{(X,j•• )};.Jisafamilyofbased G' spacClll,lhcnthewedge

V

Xjisthe

it)

quotientG-s pacelor medfromthe topologicalsum

U

Xjbyid e ntifyingallthe

j,)

basepclnta,and usingtheidentifiedpoin~asbiUl:poinlforthewedge.

3.G·adjundion squares andG-adjunctionspaces Adiagram

or

C-spac es andG-maps

willbecalledaG·a,diundionsquucifthe followinguni\'malproperlyis,."t·

isficd:

ifZ'is anyG-sp aceand

i :

Y_ Z',k:X ...Z'areG·ma~such thatj0

f

=k0i,thenthereexist! aun iquea·m aph:Z...Z' suchthatheI::::jandhoi

=

k.

For thell:iven map.

f

andithe G-spaceZaboveis uniqueup toG.h omeomorphillm inthe followingKOSC.IfZ' isany othera-spa cesuch that

isalsoa a-adjunction square,thenthereare a-mapsh:Z-.Z'andh' :

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13

z'

-+Zdeter mi nedbytheuniversal property fortheG-adju nctionsqua res involvingZandZ',respectively,whicharehomeomor phicinverses

o r

each othee. Thus,hi

=",

hI=

1',

h'l'

=

I,andh'l'

= J.

^Theuniqueness of theuniversalproperty, fora-adjunctionsqua resensuresthath'h=lz and hh'=lz',

IfAis aC-subspaceofX.thengiven the G-m:l.p

f :

A _Ywecan form the G-lUIju nd ionsqua re

A

.L;

y

x I

^X^UJY

Here,X VJYis the quotientspaceofXli Yformedbyidenti fyinga with j{a)for allacA,and giving XUIYthe G-finalst ructurewith respect to the inclusionI :Y- IYVJYandthe function

1 :

X-+YUj Ydefined by J(r )=(x}forXi X.Theactio nofGonXU, Yisgivenby

¢,Gx(X UJY) ~ XU/Y .(g,[, D

=

[gzlfo,g, G, "X

¢(g,

[, D =

[g,)ferg, G,

,<Ie

We secthat¢isconti nuous asin theproof

o r

Propositi on104.TheG-space X UJY will be calledthe G-adjunct ionspacedetermine dbyI.The corre- spond ing functi onp:XliY--fXUJYis defined in the obviousway.

1.5Prop osit ion . Thesquareusedinconstructi nganyG·adjunctio n space is a G·adju nctionsquare.

E,mcl.IfZis a G-spa ceandj:Y ...Z, k :X--fZare G·ma pssuchtha~

j0

f =

k0ithenwecan define A:XUJY--fZbyh([/( z)]

=

k(x)and A(fy!)

=

j(y)(orallxcX,yc}'.Itiseasily seenthathis unique.•

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4.Proper tiesofG·ad jund ionsquares

Thefollowin~resultswillbeutilizedinCh~ptn3onG·CW-complexcs.

1.6Proposition.IfA,8andCareG-spaces, withAaG,slIbsp.,ccofDant!

n

^IIG-subspaceofC, lhcn thefollowingdia~ ramis;\G-.djunctionsquare

C-;;;-CI A

wherei a.ndi'are theinclusions andpandp'IIrcthecorrespo nding quolienl G-mtJ.ps.

£..m2I:

IfZisaG-spaceandj:EIA- 0Z,k: C _ ZArCG'lnap~with jop

=

ko ithenWecandefinea uniqueG·map II:CIA~Zby hUeD=k(e) i£ e

rt

AandhUo])=i(A )if^Cl(A^. Then hoi'

= i

andh

oy

=k.The surjcctivityo£

y

impliesthathisunique.Finally,hisaG-mapsinceCIA hastheG-finalstru cturewithrespecttoi'and

p.•

1.7Corollary.HJis anindexingsetandforcadij(J,(Xj.Aj)isapalrof basedG-spaceswithA.;aG-subspace ofXi>thenthefollowingdiagr.,m isa G-acljunction square:

Uj,JAj

~

^Vj,lA^j

u "",j v",; ,j

whcre PA andpxarc theobviousquotient G-mApsAndij :Aj....Xiistill:

inclusion foreachj(J,end

Ui

jandVj<J i_jarc theob vious_G'lRilJl~.

j<J

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IS 1.8Proposition.Let(Xi .A';t.)be apairof baseda·spaces foreachj(Jand let{Ii:Aj^{- 0}ij};<Jbe~familyof baseda·ma ps.Thenthefollowingis a G-adjunctio nsqu are:

V'ol!' Vj<JAj -

v",;, j

Vj<Ji'j

IV",;,

Vic}Xj- Vj.J{Xi U'JYj)

V,.,"

f.m2[: Givenbased G-ma psi :

VY,

^-+ Z andk:

V

Xj- IZ suchthat

i.J ie;

e (V Ii)

⁼^k

(V ^i,i),

then for each restriction

e, :

^lj^-+Z andkj:

x,

^-+Z

I'}

,. J

thereexist saa-maphj :XjUJ,Y;-+Z suchthathjo1;=ljandhj oi;=kj.

Thenifwe defineh:V(XjUj,lj)-+Z by hl(X;U/,lj)::::lhjfo r eachjd, j<J

we havehoVlj =l,ho

Vh

=k. •

j l } joJ

1.9Proposition.LetX,YandZbe G·s paccs,withAaa-subspaceofX.

Thenif

f :

A-+ Y,andg:Y-+Zare G·maps,the outerperimeterofthe followingdiagra m is aa-adjunctionsquareforj :A...Xandthe G·map

sl.

A L y .z; Z

ij

!

I

) 1

x

7

xU/Y

T

(XU/Y)U,Z

CI2cl.

LetI:Z- Wandk:X-+WbeG·maps suchthat

to

(gJ )

=

k0i.

Thcueln cc theleft-hand sq uareis a O·adj unct ionsquare,there exish aG- map h:XU,i'- IW suchthalhOI=log andb»]=k, Butthe right-h andsqu are is alsoaG-adjunctionsqua resothe reexists aG-m aph':(XU/Y)UJZ_ lV suc h thatII' o ~⁼tandh'og=h.Henceh'og0

J

⁼hoj=k and so the outer perimeter isaG·adj u nctionsquare.•

(25)

16

5. Comap? ingcylinder

The a-mappingcrlindcr forthe G-mapf:.\'-+ Y is definedallthe a-adjunctio nspacefo rm edviathediagram

x ,

^{OJ

-l....

^l'

,I I,

Wewilldenote(Xx1)U,YbyM,andcallittheG·mapllillgcylinderfo r

f.

Theact io n ofGallX xIisthe diago nalactionwithGacting trivially on I.H(X,. )and(Y,. )are based G·spaccsand

f :

X ...}' is a baseda-map then the~G-mapping cyli nder,denoted by,fll,is th equotie ntG-spare Ai }=~,with {.X1)asbasepoint

Let

f :

X_ Yhe a basedG·mapand

,.i

j be thereduced G-mapping cylinde r. Then we havethefollo wingcom mutati ve diagram:

x ...!... y

whereiand r arc the based G-mapsdefinedby

;,(X, ') (Ai" .) (.:,OJ,and r, (Ai" . ) (Y,.)

[yJ~ y [x,/1 I(x), whereId,uXandy"Y.

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17

We also havethe based G·mapj:(Y,.)--t(M" . )definedby y1--+!y].By an argument a.nalogous to tha.t innon-equivariantcase,r is a G-bomotopy equivalencewith inverse[,

6.Expandingsequence or G-spaces

LetXo CXlC•.. C

x ..

C.••hean expanding sequence or G-spaces,i.e.

X.. is a G·5ubspar.eofXnH foralln~0,and the setX

= U

X".Let ,,;::0 p:

U

X" ... Xbe definedbyp(z)

=

inez) whereZf Xn^andill:X.. ...Xbe

n<?;D

the inclusion.So,by Proposition1.4,.Ycan begiventhe C-Iinalstructure withresp ectto theinclusio ns{in:X..- X}II~O'From theuniversalproperty, it fellows that

f :

X ... Z isaG-mapifand onlyi£IIX"is a G-map foreach n~O.

1. 10 Proposition.IfXo^{C XI}^C^•.^•^CX"

c ...

CXis an expanding sequence or a·spaceswhereXhas thea·finalstru ctu rewithrespectto thcinclusions

{inX ..<-tX}n;::o,andAis locallycompact Hausdorff,then

Xo)(ACX_{I )(}A

c ...

X..

x

A

c ...

CX

x

A

isalso an expandingseq uenceofG-spaceswithXxAha ving the C-final structurewithrespecttothe inclusions{i..x1,, :X..xA~Xx A)..zo.

.f.m2{:Leth:X x A-+Z bea functionfromX x Ato the G-spaceZ such that IiIX.. xAisaG-map foralln2:O.By the exponentia l law forG-spaces andG-maps, foreachhJX.. xA there existsaG-maph':X ..-+Ma po( A,Z) and:-.functionh:X-+ MaPG(A,Z)suchthath'=hIX...Since X has the G-finalstructurewithresp ecttothe inclusions,itfollows that

h

is aG-map and hen cehis alsoa G-map.Th usXxAhas the G-finalstruct u re with respectto theinclusions{in)(A:X ..xA4 XxA},,~o.•

7.~

LetXbe a based G-space.ThensinceXx

to )

U• xIis;>a-subs paceof

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18

x

^X1,C{X)

= x.MS .•,

^isa quot ientG-space.Itwillbe calledtheW1£of X.

DefineS;'=

~

ttx

*

^and^Eir⁼

~.

l1)C . ^where

*

^x^{S" and}

^~

^x^E"^have

thediagonal act ionwit hGactin gtriviallyonS"andEn.The followingfact isneededintheproofof theDrown Ilcprcscntablllty Theorem:

That the lefthand~above isa homeomor phism follows from thefo llowing.

1.11 Proposit ion . LetX andYbe baseda·spaces,where the underlying space ofXis locally co mpact Hausdorff. Then there isa homcomorphistu

(X x Y) XxCy .

!{J:C ~ -+

---xx;-

definedbythe rulethatthe equiv alenceclasses of(z,y, t ),in thetwospac es,corr espond,whereXfX,y'Y .tel,

. . . r

h

C (8)

⁸^<I

~.RccalhngthatIfAISasub sp ace081 en

A

a(By.{O})U(A x1)'

. (X

^X

Y) X

^x^Y^X

I

we nctfcetho'llC XX; a(XXYX{O}) U( Xx {.}x

t

WealsonotethatXx

(~)

is homcomvrphi ctothe quotient spnceof XXBin which{e] xA,foreachZfX,isidentifiedt.oa scparetepoint.lienee Xx CYis homeomorphicto thequotien~space ofXxYxIwhereforea chzeX, ({z}x Yx{O} U{:e}x{.)x/ ) is identificdtcaseparatc polnt,SinceXXXxG:Y is aquotient space ofXxCYandCYis a quotientspace ofYxJ,then, usingIBroJ,4.3.2,p.101,

x.;e:-

is (essentia lly)a quctlcntspace ofXxYxI under the equi valencerelat ionwewilldeno teby...Thenwehave(:e,y,O)·...

(z,.,O)...(.\J ~ ~~o~uX,yeY.all~(%,.,/)...(%'X~Y~.t,O)for zeX,

«:

Hence----xx;.-J9 homcomorp hlcIo(XxYx {O})U(Xx {.} xI)' In bothcases the actionsofGagree with lhat"IIthe quotienta·space

XxYxl ,. lll ducedb hcdl I ·

(XxYx{OllU(Xx{.}X/ ,thatJsJt se muuceu ytCUJagon aecucn ofO onX xY xi .

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19 Using p, q,rand sto denote the quot ientmaps:

XxY

:(~)X I~C(£>:.!:) .

p:X xy...~, q

x

«, Xx-

r:}' xI_CY and.,:XxC}"_XXxxC,Y, we note that we have

(X

XY)

q(pxl,):XxYxl_C

Xx.

^and

,,(Ix^{x r}:}^XxYx1_\ XxC,Y .

SinceIamiXarclocally compactHa usdorff,q{px1,)and"(Ixxr) arc identifications,T/Jisabijcctlcn, and since,pq(pxii)=J(lxx r)itfollows that",isII.homeomor phism.•

S. Suspension

LetXbe a G-spacc with basepoint '.ThenXx {OJUXx{I}

u.

x1is a G-subspace ofXx 1 and the suspensionofXis thequotient G-spacedefined by

S XxI

X= Xx(O)UXX{l} UoxI 9. Smashedproduct

LetXandYbe bascdG-spaces. ThenXVY=(Xx{.}) u ({.}xY)isa a·subspaceofXxYand the G·smashcd product isthe quotienta·space

XAY=~~~

Fillally,let

p',,)

be a b;UM G-space andXis a G-space.We define theG·

space

x:

by adjoiningtoXan additionalpoint00onwhichGact s trivially.

XxV Thenwe haveXi-IIY==,~.

Ifl/is a closed subgrou pof G andXisaG-spacc,we defineXI(={u X /hz

=

;rfor allh}withthe subspace lopology . Then, ifXis Hausdorff,XIlisclosedinX

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20

(sec(OJ.3.9,p,25).If

f :

X ... }'is aa.ma pthen /":XII-+ylldenotes the restri ct ion offtoXH.

1.12Proposi tion.IfX andY arcHa usdorff a·spaces, thentheadjunct ion space X^I1U/Hyllisaclosed subs paceof XUJ Y.

b.29I.SinceAllCX",X" is closed inXandyllis closedin Y,it follows fro m a result of gene raltopology (sec[Dul.VI,6.4, p,128) thatX"U",yllwillbell.closed subapece inXU} Y••

Fromthisitfollows that sinceXIfU/Hylland (XU!y)/1havethe sameunderl ying

~,and (XUJ Y)"isasubspace ofXU,Y,they mustbe homeomo rphic ,l.c.

X"U/Hyll=(XUJY)u.

In orderto show that the G· CW-complcxcs defined in Chapter3 arc llausdorff weneedthefollowingresultabout adju nctionspaces.

1.13 Proposition.IfXandYarc norma lspaces andAis a closedsubsetofXthen for any map

J :

A-+Y,XU,Yis alsonormal.

.P.m2(.See{FPj, PrcposltlcnA.4.8(iv),p.260.•

1.14 Proposi tion.(i)IfXandYarc G-spa.ces then(XxY)1l=XIIXl"JI.

(ii)If{Xjlie:J}isan indexedIamily er based G·spaces,then

(V

Xj) H=

V(XJ!l .

~'J

^j.J fiQQf.Both (i)and(ii) followquite easily(romthe definitionoftileactionsofGOIL the spacesXxYand

V

Xjorespectively.•

j<J

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21

Chapter2Equivariant Homotopy Groups

Two G-mapsf, g:X- IYaresaid to be a-homotopicifthere exists a G-map F:X xl.... Y, withGactingtriviallyon[,auch thatforallztX,F(x,O)

=

fez) aut!F(z,l)=g(x). Notethat each map PI:X _ Y defined by Fj(x)=F(z,t)is itselfaa-mapsince foruX, geC we have Fj(g%)=F(gx,t)=gF(x,t)=gFI(x).

The set of G-homotoPiclassesof G-maps fromXtoYwill bedenotedby[X : Y]a.

IfA is a a-subspaceofX andB is a G-subspace ofY then two G-maps 1,9: (X, A) ... (Y,B)arc said to bea-homotopic~ifthere is a G-mapF: (XxI,Ax/) ...(Y,B)such that forXfX, F(x,O)=f(x)andF(z,!)

=

9(Z)and for"cA,F(a,l)

=

I(a) for tel. The set ofrelative a·homotopyclasses of G-maps (rom(X , A)to(Y, B)is denotedby[(X,A) :(Y,B)Ja.

IfXandYarcbased G-spaces,thenthe basedG-maps

Ls :

X_ Yare ba~cdG-homotopicif there exists a G-homotopyF : XXI-.Y between / and 9 withF(.,l)==

*

fortel,We write /':::!.G9 todenotethe based G-homotopy.The setof based G·homotopyclasses of based G·mapsfrom X toY willbe denotedby [X,YJ~.'[(X,. ) ,(Y")I~.

Let(X,.)be a basedG·space.Thenfor each closed subgroupHof G, the under- lyingset of the nth e9uh:arianthomotopygroupof typeHis definedby

1r::(X,*) [SH:X}~

~ [ (W

^X^5",

w x.) '(X" ll:

the canonicalbijection~being defined usingthe universalproperty of quotientG- Qx

S" (G)+

spaces , ...hereSil=_~e

11

1\Sr>.Using the fact_~l~alA1\5BSiS(A1\B) wehnve H

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22

SinceSiiisit.suspension forn~1,...ecandefinea groupop«ati on on ...!'(X••)ina way analogoustothenon-cquin.rianlusc.1f

Ul.

(gJn~'(X,.)aret,,"Oelements ,then theprodu ct of[J]and

[gl .

denoted.(J).[g).istheG·homoto py clastofthe composite G-map

Sj,2..t SYVSj,!:!!.XVX.:!. X

whereVisthefoldingmapdefined byV(x,.)

=

^{x, V(}^.,r)

=

^{z for}^alln.\",anti111/

isthecomultip licat ion G-mapgivenby:

lin:5i/ 5

1 /

VS'H

{

([g1l,[=,21J1.-) if O:5l

:5 1.

%,5..-¹

[gH,I ' ,'lJ

h [gIl,(z,21-1)])if 1:::;:1$1, z(5..-¹ Thisuses the ractthatSit""

(i )+

^AS'^=-

o r

^1\^{5(5 ..-}¹⁾^{so that}^an^{cleme nt}^of

Sitistheequivalencecl~(gIl,I=,tJlwheregeO,US"-Iandtf!.'Then1111is ca,ily K'CntobeaG-map.

2.1Proposition.LetIIbeaclosedsubgrou p and(X,_) be" basedG-spa ce.There existsa na t ural isomorph ism

Toprovethispropos itionwe needthefollowinglemmas.Firstlet MaPa(X,Y) denotethesetofG-mapsfromXtoYand Map(X,Y)denoteth e set ofallrnaps fromXto Y.

2.2I&mm..a..IfXand Yare G-spaccs,withGactinglriviallyon thelocallycom pact Hausdorff spaceX,thenforeach closedsubgroupJI

o r

Gthere isabijection

Mapa

(~

⁾⁽^X,y)

~

^Map^(X,^yll^).

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23 f.r22f;Deline0 :Ma pc (~xX, V)-+ Map(X , yll)byO(F)(x)

=

F(H,z)for r«Mapa

(¥,

xX,V)andzeX,and defineif!:Ma p

eX,

yH)-+Mapc

(ft

xX,

v)

by~(f)(gll.x)

=

gf (x)for[cMa p(X,ylf).Wefirst show0and

9

^arewell-defined andcontinuous.

SincehO(P)(x)=hF(Il,x)

=

F(hIJ,x)

=

F(Il,z)=O(F)(z),it followsthat the image ofO(F)is a subsetofyll.Definingi:X-+~xXbyi(z )=(/l ,Z),where :reX, wesee thatOfF)=Fi,so0l~:')iscontinuousand0 is well-defined.

We noticethatiC9h 92CGand 9'2'9 1(8, l.e. 9'=92hfor somehell,thenoil(r)= 91111(x)=911(x) (or allxtX.Hencer$(f )is well-defined.Thefunction m:GxX-+

X,definedby m(g,r)=9f(r )where9(Cand;teX tisclearlycontinuoussince q xIx:GxX-+

ft

xXis an identification.Now¢J(j)(q x Ix) =mso¢J(f) is continuous. Also,g'4JU)(gll,z)=9'9[ (Z)=,p(f)(g'gll,z )=tjl(J )(g'(gll ,z)) so thatq,(J) is aG·map and,pisa well-definedfunction.To establishthe biject ionwe have lo showthattP00 and 0o,pequal the respectiveidentityCunct ions.

LetF£ Mapa

(»

xX,Y).Then

(ooO)(F)(gll,x)

=

o(O{F))(gH, x)

=

gO(F)(x) gF(Il,x)

=

F(gll,x) So (,poO)(F)

=

F.Nowlet[cMap(X,ylf ).Then

(00o)(/)(x) O(o(/»)(x ) oU)(II,x) fix)

Helice(00,p)(f)=I.•

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"

2.3

1.srD.wA.

U Xand Yare baseda-spaceswith G actingtrivially on the locally compact Hausdorff spaceX.thenIor cechclosedsubgroup11ofGthereis.\ bijection:

~:We define0and¢as in Lemma 2.2, notingthatO(F)(.)=F(ll,.}

= .

^aud

'U)(g1l,·)~gf(.)~g.

=•.

So 0 andrPtake based maps to based maps, which allows us to establishthe biject ion between thetwo sets. •

2.4I&mm..a..IfXandYarea-spaces withGactingtrivially on the locally compact Hausdorff spaceX,Ais any G-subspace ofXandZis any G-subspace of }',with .fAcXand*fZCYthenforeach closed subgroup1lofGthere is a bijcctlou

. (G . G G)

f.rQQI:Define0and rP asInLemma 2.2.LetFe l\Iapo

Ti)(

J.:,

71

XA,

Ii

x.

andfd.1ap(X,A,*), (Y",Zll,*» . IfacAthenO(F)(a)= F(J/,a)cZ"since hF(ll,a)

=

F{hll,a)=F(JI,a)for allhdl,and¢(f)(glf,a)=gf(a),Zsince f(a)cZandGZ==Z. Since0and

,p

are inverses or eachother,thi3allowsU9to establish the bijection .•

IfXisreplaced byXx /andAby •x 1inLemma 2.4,then we havethebijection l\IaPG

((*

x

X

xI,»X. X/) :

(Y ,.»)

~^Map

«X

x1,•x1) :(Y1l,.)).

So there is a bijectionbetween the based a-homotopicsor

~

_Il_{x e}^to^Y^{and Lased}

homotopiesor X toyJl. Prooror Proposition 2.1

Let0:Map~(Sif'X)-+ Mapo(S",XII)be defined by o(J)(z)=f(l/,::)for

as-,

Then byLemma 2.3there is a bijection

(34)

s s

Q.:ISii:XJ~

...

(5":

xlif

wherea,([JI)=[0(/)1.We have to show~hatc,isa homomorphism.Let

/0.11 :

51,-+X.ThenQ.is a homomorphismif

a.(lfol ·lId) =

Ci.((/OJ) ·O.([!II).Let 0(/0)=90and0(/1)

=

9•.Then

Ifol ' Ud

isthea-homotopyclass ofthe composite

SH ..!:!!... S'ir ~ XV X -.!.. X.

Similarly[gol[gIl is the G-homotopyclass of thecomposite

lienee a.willbea homomorphismifwe have

cr.([V'o{/oV

Id

0I'll])

=

^['VIII⁰(goV91)011].

Themap"(11/0(gOV91)0IIis given by:

{

([,.2'1.') ~ g,([, .21]) 0';'';

I.

,,8'-' h(=,21-1]) 1---+ 91((Z,2t-1)) ~.:5:1:5: I,%(5..-1• The map V0(foVII)0V/Iis givenby:

{

([gll.[, .2'i1. ') ~;f,([gH.I,.2'iJ) 0';' ';

I. ,,8'-'

(••[gll.(,.2'- 11lJ ~ f,([gH., .2' - 11IJ

I,; ',;

I.,,8'-', If

f :

S'H-+Xthena.([J])isthe G·homolopyclass of the composite

(35)

"

s..!.!!...S;i L x

lIenceo.([Vo(/0 v

I.)

0

vll11

is theC-homotopyclass of

SPI S;rV SH - X

{ ([JI,[z,21J1,') (',IJI,[,,2'-IJI)

fo([JI,I,,2'iJ) f,([II,I, ,2'-IJl)

O:5!:5~

~:Sf::; !.

B,'f,([Il,[,,2'iJ)

=

go([z,2<]) and1>(11I,[z,2' -IJI)~g,([z,2'-IJ), whichimplies that0.([\70(faV III0II,,])=

iV"

0(goV91)0c].Thus o,i,a homomorphismand consequentlyanisomorphism.•

Let(X, A,. )be a pairofa-spaceswithbasepointe, Thenfor each closed subgroup HofGthe nthrela tiveequivarianthomotopy groupof typeJlis definedby;

1t~(X,A ,.) [(Ei't,Si,-t,.):(X,A, .)J~

~ {(~ x

^E",

~ x S\ ~ x.):

^(X,^{A, .) J:}^.

2.5Proposition.Thereis a naturalismorphism

£rW:

This follows from Lemma 2.4, with the proofbeing similartotile proof of Proposition2.1.•

2.6 ~. Givena G-map0' ;(EH,S}',- I,.) -+(X , A,* ),then(01

=

0in 1f::(X, A, * )ifand onlyif^Qis G·homotopicrelativeto5;'-1toaG·mapBit--tA.

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27

f.rllill:

FromProposition2.5itfollows that[e]=0in~~'(X,A,.)ifand onlyif forthe correspondingmapa' :(E",S,,-I,.)-+(XII,AI/,.),wc have[0']=0 in r ..(XJ1,AII,.). Then a' is homotopicrelativeto$"-1to a mapEn_ Allbya homot opy

F':(£",S..-I,.)x1_(X",AI/ ,.)

(sec for example(Sp], Thm.1,p.372.)

Also aa-homotopyfromQto a G·mapEit-+Ais givenby

F:

(% x

^En,

%

^X^5..-¹^,

* ^x.)

^X¹^-+^(X,A,.)

whereF(gll,=,t)=gF'(:,t) for geC,leE"andIll.

This G·!Lomotopyisrelative to5;,-1since for zo,5..-¹we haveforallttland9'C:

F1(911,Zo) 9F:(Zo) go'(:o) 90(H,:0) 0(9 H, zo).

Conversely,assumethat thcrcisaG·homotopyF:

(*

^xEn,~X$"-1,»X .) X J-+(X, A,.)relative to S'H-^twithFa

=

0:andFa(En)f; A.Then[c]=[FII in lr~/ (X.A ,.)and a G-homotopyJ((rom PI to the constantmap» xE"_*fXis givenby

whereK(gJl,%,I)=F1(glf,( 1-l)=+tt). •

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28

Letn~O. Apair(X.At.)of basedGspace!is saidtohe~ifevery pathcomponent ofXintersectsAand,Cor15k::;n, we have"'l' (X , A•• )

=

0for eachclosedsubgroupHof G.Asa directconsequenceofTheorem2.6 wehavethe followingcorollary.

2.7 Corollary:A pair(X,A,.)isG·n-oonnectedforn~0ifand onlyiffor 0 Sk:5II every G·ma pQ:(£ 1,8;'- 1,. ) - (X,A ••)isG-homotopicrelat iveto

st t

^lto some G-map

E;"_A.

Finallywenotethatfromthenatur ality of theisomorphisms1r~1(X ,...)~11"..(XII ,.) andlI"~(X,A,*) ~If..(X H,A^H,*)we havethatfor eachclosed subgro upJI

o r

^Glhl~

{allowingdiagramcommu tes

--+ ...

,

7r..(XH,A", *)

...!...

",,,(A" ,.) -!'. 1r..(XII,. ) ....!:....1f..(X l/tAII , .)

..!... ...

Since thebotto m row isexact andthe verticalhomomorp hismsarcin fact isomorphisms,itfollowsthat thetoprowisalso exact.

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"

Chapter 3 G·CW-Complexcsand the EquivariantWhiteheadTheorem LetnbeIt.fixed positiveinteger.The G·spaccX is saidto be ob tainedfrom tileG-spaceAbyatt aching cquivariantn-cellsifforIt.familyorclosed subgroups {llilid}ofGwe haveforeachjlJan indexingsetLjand for each).f Ljthere is a copy(E.\',Si-^I)of the pairof triviala·spaces(£ " , S..-I),and we have the following G-adjunction space:

Jj ff:-

^{X E.\'}

"',

We denote

tPl*

xS;-Iby

tPl

and

¢ If!;

x

EX

by

¢t.

We call

¢{

the attaching map for the cquivariantn-cell

7h

^x^E

A

and we call

3{

the characteristic

map.

We say thatthe pair(X, A)is a G-adjunctionofc9uivariantn-cells.

3.1

Lcmm.a.

If,inthe situationjustdescribed,Ais normaland Til thenXis normal andTlo

£mill .

^SinceHjis closed inGfor eachjd,itfollows (secp.3)that

fi;

is Hausdorff for eachicl,SinceGis compact

fi;

is compact, and hence

fi;

^xE~isa.compact Hausdorff G·space for eachjfJ,),f Lj•Thus

if;

x

Ei

is norma lsince every compact Hausdorff spaceis normal(sec [Hu], Proposi tion2.7, p. 63).Hencethe topological slim

H if;

^xEiis normal andby Proposition 1.13,Xis normal.AlsoXis T₁since

".L,

bothAand

Y fi;

xEiare T1(sec[Hul,Proposition 3.7,p,125).•

"ILl

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30

Let(A,")bea.based G-Spll.CC, and let{/lilj(J }be afamily ofdosed subgroups ofG.Iffor eachieJ there is anindexingset Lit andfor each),cLjthereis a copy (Ejf,,.\.Sj,~.Uof

(Ei't"

Si/; I),and¢:~

S it.\ - A

isabased G-map,then there is

~,L,

a based G-adjunctio nsquare:

We saythat the pair(X,A,. )is abaseda-adjunct ionofbased CQuiva riant n-cclls, orthatXisobta inedfromAby attachingbased cquivariantn-cclls.

AO-CW'complexorsimplyG-complexis a G-spaceXsuchthatthere is an expan di ngsequence ofa·spaces

XoCXlC ...CX"

c ...

withthefollowingproperties

(i)thereis a familyof closed subgroupsHjofG indexedbyJ,l.e.{lIj

li cJ},

withthe property that foreachjtJthereillanassocia te dinde xingset Lit and

X '= !J ,. J (if:)

^J x

P l,

~.LJ

(ii)(X",X"-l)is a G-adjunct ionofequivarlan t a-cells, and

(iii)X hasunderlying setUn~oXnand carries the G·finalstructu re withrespect totheinclusions{in:X" ... X}..zc-

The~inwci2n.ofXis the largestnsuchthatXcontains ancquivarian t n-ccll.If no suchn existsthen thedimensionofXis saidtobe infinite .

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31

Moregenerally,arda tiveG-CW·complexi5defined as follows.IfAisanyG·

space,then(X,A)is ard alh-eG-CW· complexifthere eJ:ists anexpand ingsequence of a·spaces(notpai rsofG·spaces):

A

=

(X ,A r '

c

(X,A)'

c

(X,A)'

c ...

C(X, A)"C... wil h thefollowingproperties:

(i)there is anindexed family{Jljlid }of closedsubgroupsofG,suchtha.l for eachjcJthere is an associated indexingsetLioand(X, A)O=All

(U (11;)

x

(AI).

j<J

~.LI

(ii)((X,A)", (X,A)"- l )isa a·adjunctionofcquivariantn-cclls forn2::O. (iii )X11Munderlying set U..~o(X, A)"and carriesthe G·final struct urewith rc-

sped tc theindusions{i..:(X,A)" ...(X,A)} ..~_I'

The relati ve djm" m jou ofeX, A),writt en dim(X -A),isthelargestvalu e ofn suchthattheconst ruct ionofX fromA includesancquivarian l a-cell.J(nosuchn exists,therelativedimension isinfinite.

AbasedG-CW-co mplexcanbedefined as abased G·spaceXbytaking thedefini- tionor G·CW ·com plcx,requiringXutobea I-pointspa.ce.,replacing G·spacesby basedG-spa c.cs,mapsbybasedG·maps, disjointunionsbywedgesand G-adj undions bybasedG-adjunctions.

Similarly,a basedrcl&tive C-CW-complex canbedefinedas abasedG-spaceXby taking thedefinitionorrelati ve G-CW -complexandreplacingG·spacebybasedG-

IJl ;\CC,C-ma pbybasedC-map,disjoint unionbywedge, and G-adjunetionby based G·M1junet ion.

3.2 Propositi on.Aa·eW·complexis norma landT1(and hen ce Hausdo rff).

£.mill.

Since XO isdiscrete,itis norma l andT1•ByapplyingLemma3.1,we seetha t eachX..isnormal andTlfor eachn~O.SinceX=U"l!;uX ", itis a normalG·space . Hz isa point orX,thenZ(X"forsome n~

o.

ButX"isTl ,10that{e}is closed

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32

inX".SinceXⁿisclosed inX,itfollows that[e ] is alsoclosed inX.Hence,Xis Tlo .

Sinceany connectedG·CW.comp!cxis G-homotop ycqui\'alcnttoa.based

a·cw·

complex(see [FPI.Cor.2.6.10,p. 82 forthe ncn-equivariantversion],itsullicea to consideronly based G·CW-complcxcsin our discussion.

Intact, each basedG·CW.complexisalso anordinary G-CW.complcxifwe neglect basepoints.This easily follows fromthe followingresult,

3.3L!:mm.a.Let(A,.)be a basedG-spaceand(X, A,.)bea basedG-adj unction of based cquivariantn-cells.Then, disregardingthe basepoint,(X,A)isaa·adjunction ofequivari antn-cells.

f.rQQf.Considerthefollowing diagra m:

LJ fI;

X 5;-1

V f!;xSl'-t V

Si,~~1

...t.

A

",

i^{t J} ^j<J

.l.L, .l.L, ),.L,

I

JJ ^fi;

^xEl'

^V

jeJ

^ftxEi

I it)

^V

^Eir,). ^~ ^X

>.,L. )"L , .l.L ,

The right-hand squareis a based G-adjunction squaresince(X,A,t-)is a based G-ad j unct ionof based equivarlanta-cells. The middle..quare is a a-adj ullct ion square byPropositions 1.6 and l.B.Tbleft-hand squarcis a a-adjunction square by CoroUary l.7.Finally,byProposition1.9,the outer perimeterofthe above diagram is a G-adj unctionsquare. The resultfollows.•

3.4~.If(X,_)is a based G·CW-complextbcnXis a G-CW-complex.

~.Thisfollowseasilyfrom Lemma 3.3andthe relevantdefinit ions.• Fromthiswe can concludethatresultsgiven below for ordinaryG-CW.complexcs [e.g.Propositions 3.5,3.6 and 3.9) alsoholdfor based a-CW-oomplcxcs.

The a-mapi:A _ Xis called a~if for all a-homotopic,J(: AxJ _Yand a·maps

I :

X - Ysuchthat

Ii =

[(G,thereexists a G-homotopy

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"

F:XxI _ YsuchthatFa

= f

andiji

=

X, for0$r:51.Thisis equivaleat to sayingthatthefollowingcommutativediagram canhe completedbymansofl-

e-mapF:Xxl _Y.

/ r~

Ax{OJ XxI----W

~

_X_x

J~

_{O} _f

IfIiisa G-subspaceo!X,thenAisca.Ileda~ orXilthereexletsaC-map r :X-+Asuch thatriAistheidentity. TheC-map ris called a~ofX cntcA.

3.5Proposition.ITXis obta ined{romthea·spaceAby attachingequivaria.ota-cells,

thenXx

to}

uAx/isI.e- retract

oe x

xl. .

fm(:We know that thereexislsa non-equivariaatretractionp:E"xI-+E" x

to }

U$"-1XJ(seeforexampleISp),.3.2.4, p.117).ThenifHisaclosed subgroup ofGan equivariantretractionisgivenby:

lx p: ix E" X]

(gH,z ,' )

*

^x^E"

^x ^{{O} U ~x}

^{5"- 1}^X^I

(gH,p(z,<)).

Let~:

if;

x

E! -

X bethe characteristicmapslortheequivatian ta-cells

fi;

xSitjll,),tLofX.Thena. G·retractioo can bedefinedby:

R:X x/ _ Xx{O}U Ax I

(=,1)~ (z,I) ,(¢{(gH;,=l,t ) ~ .\«1xp)(gH;,z,1)) where%fA.,

«t,

gHjf n;Gand:(X.•

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3.6 Corollary.IfXis obtained from the G-spaceAbyatt aching equivariantn-cetts, then the inclusioni:A ...Xis11G-cofib ration.

E.r2Q!:LetR:X xI _ X x{OJ UAxIbe the G-retractiongiven bythe above proposition .We have to show that the followingdiagramcan be completed :

...-"A i l~

AX {O} Xxl-- -.,.Y

"'-.x J{O)~

whereYis anyGspace,[{ and

f

are anyG-maps.Thiscan be done by lettingF bethecomposite

XXI~XxOUAxl~Y.

Thediagramwillbecommuta tive sinceRIAxI=AxIandIlIXx 0=X x O.• Wecan extend thisresult to the casewhere(X , A)isa relative G-CW-complcx.

3.7 Proposition. Let(X,A)be a relativeG·CW-complex.Then the inclusionj ; A ...XisIIG-cofihration.

£IQQ{;LetJ(:Ax1_Yhe a G-homotopyand

t .

Xx{OJ _YbeaG-map wit h f1Ax{OJ

=

^/(0'We will constructa G-homotopyF:Xx 1-+Yind uctively on the n-skeletonaof(X,A).

We have

A=(X,Aj-'

c

(X,A)'

c

(X,A)'

c ...c

(X,A)"C...CX where(X ,A)kis obtained(rom(X,A)l'-' by attaching equivariant,(,;·cclls.

Consider the followingdiagram :

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The G·homotopy£0existssince(X,A)Oisobtai nedfromAbyatt aching D-cells, so bytheprevious corollaryA....(X,A)OisaG·cofibratioll.

Ingeneral...ehave:

~ "(,AT

^x

1 :;:-1

(X,A)'-'x(O) (X,At xI...!...,Y

"'-.. J /":

^{/eX, A)n}^'- ⁰

(X ,A) x(O)

Hencewe can construct asequenceofG·homotopie:s{F" :0~-1)withF-l

=

^A'

suchIhat:

(;) F , =

f1(X,A)"

(;;)F"!( X,A)"-'xI)

=

F' - '

DefineF:XXJ-+YbyF(z,t)=Fn(z, t)for%((X,A)",tel,Fiswell-defined sinceF"!«X,A)n-1x1)=po-I50thalif%( X,A)O,F"(z,t)=pn-l (z,t)

=...

⁼

F-I(z ,t)=K(z , t).Wca1soknowthatFisa G-map sinceFI«X,A)"x l)=F"and (K, A)xJhasthea·finalstructurewithrespect to{(X , A)·x1}"~_1(byLemma.

3.3).

Finally Po

=

I,sincc b)'(i)if%(X ,A )",

F,(r)=F;'(r)=fer), andFl."x

't

=K,sinccifz(Athen

F(r,l)=r '(r, l)=K(r,t), whichimplies thati:A~XisaG-cafibration.•

a.s

Lcumn.

FcranyIamily{X;}jd oftopo!ogica! spaccs, =-.

( v( X ;

^x^£"+1⁾^,V(Xjx

sn») =

jd jd

o

for0<k:$II.

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"

fI22C:Weusethefactthat

V e x,)(

£,,+1)i:!

(v X i )

^XE"+1ard

veXj)(

S" ) :r

ieJ j.J i.J

( V

Xi)xS".LetX=

V

Xiandletp:XxS"- Xandq:Xx£,,+1 _ X

j.J jeJ

bethe projection maps.Then since pandqarefibr:t.tions withfibresS" and

e"u.

respcc til 'cly,we havethefollowingdiagra:n:

withthe twohorizontalrows exact.Sinceqi=pwe lmvcq.i.

=

p•.AlsolI"k(S")

=

0 lor0<k<nand1'.4: (E"+I)=0 for 0<kSn

+

1.Dyexact ness,p.andq.nrc bothisomorph ism s(Of0<I.:<n,andhence,i.isanisomorphism lor0<k<II.

Fork

=

n,q. is an isomorphism andp.is anepimorphin.: llcncei,must be.UI epimor phismfork=n.

Now considerthelongexact sequence

SinceI' .is anisomorphismfor 0<k<n andanepimorphismfork

=

11itfollows that1r!;(XxE",+I,XXS" )

=

0 for0<k::;n .•

3.9 Proposition. HXisobtainedfrom thebasedHausdorff C-spaccAbyal~nching basedequivariantn

+

l-cclle, then(X,A)is C·n-connected.

fmm;Let(X ,A,-)be a baseda-adjunc tion

o r

based equlva rlant n-cclls:

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37

j~ D;x S'J.

I.. Lj

where{J/j JjcJ}is an indexed familyofdosedsubgrou ps ofGand{Lj li d}isan indexedfamilyofsets.

'ra king J1·fixcd pointsets we secfromProp osit ions1.12and 1.14 that tbefollowing is anadj unctionspace:

V (O;)H

^XE

A+ 1 -.!.:... XH

jeJ )"Lj

Since

(~) ^H

^x

^SA

^isa deformationretractof a neighbourhood of

(fI;)H

x

EA

^+1,

andI'k

( v ( Ih)H

^X^Ei+1,

V(~)H

^X

5") =

0 for 0<

1:.:s

n (Lemma 3.8)it followsas aspecial case ofthe Blakers-Masseyexcision theorem,(see[OKPI,p.211).

that(XJr,AH)isn-connected. Hence'II",I, (X H, AH)'""0 for1

.:s

k

.:s

n andso,by Proposition2.5,I'.F(X,A)=0 for1:::;k:S: n.So(X,A)is G-n-connected.•

3.10

kmrD.a.

If Xisobtainedfrom thebased G-spaceAby attaching based equlv- arianla-cells, and(Y,B,")is a based pairof G·spacessuchthat1r!!(Y,B,.)

=

0 forall closedsubgroupsHofG,then any G-mapf:(X,A ,* )-+(Y,E,.)is based G-homotopi c relativetoAto a G-m apk:(X , . )-+(B,.).

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- - _ . - ---- - - --

- (G G ,G ~

fm2{: Let 9;:

7ij

xE"',

tii

x5"-.

Jf.

X. - (X,A, .)eorrcspcndtothe attachingmap! of the cquivui&lltn-cells.Thente composite

represents ahomotopy classinll'~"(Y,B,.).By Theorem2.6,there exists:lbased G-homoto py

such that

(i) FAgHj,z , t)

=

!(9j{glij .z))for(glfj, z}~ R;x 5"-1 (iilFj(gH;" , O)

=

/« ;(g//;., ))

(iii)Fj(gHjoz,l)t:B.

We canthus defineabasedG· homolopy

K:Xxl_ Y

by K(z,l )=f{z ) for %t:A,tel

and K{¢>j(gHj,z ),I)

=

F';{gllj,%,t) for

(gl/itz)(~)( e-,

1<1.

ClearlytheG-homotopyis,d ativeto A, and

K(4Jj(gH;,z),1)=Fj(gll" z, l )cB for (glljozkjJj x E".G

Thenk(x)=K(x,1)istherequireda-map .•

3.11 Proposition.If(X,A,.)is abasedrelef ....eG·CW·complcx

o r

relativedimension :5n,and(V,B,.)is G-n·connected,then anyG·map f:(X ,A,.)- (Y,n,.)is baseda-homot opic,relati ve toA,to a G·m apfrom(X,.)_(D,.).

(48)

39 fm!l(:We have thefo llowing sequenceoCa·spaces:A=(X,A)Dc...C(X, A)"C ... c(X,A)whcrc«X,A) k,(X,A)1-- 1)is abased G-adjunction orequivariantk-cclls.

Since(X, A)'is obtainedfrom(X, A)O=Aby a.ttachingbased equivariantl·cells,by LClIlma,3,1O,fl(X,A)'is based a·homotopicrelativeto Atoa.G-map(X ,A)l_B viaa basedG-homotopyKI :(X,A)!xI_ (Y,B).Since(X,A)!~(X,A)is a a·cafibrationwe can extendJ( 'toa based G·homotopyF¹:(X,A)x1 _(Y, B) suchthatF' !(X ,A)' xl

=

KI.

Now assume thatF""-I:(X,A)x 1 _ (Y,B)with m~2, has been constructed suchthdF,,,,-lj(X,A)"':(X,A)'"-+(Y,B)isa based G-mapwi thFj-I«X,A )"'-l)C

n.

ByLemma3.10 thereis a based G-homotopy

tcr.(X, A)"x1_(Y,B) suchthat[<j {X,A )

c

Band 1('"isrelativeto(X,A)",-t.

Since(X , A)'"^'--t(X,A)is a G·cofibrationwe can extendK'"to a basedG·

homotopy

F" ,(X,A)xl _(Y,B) suchtllalFj«X,A)"')

c

B.

lieneewe have a sequence ofbased O·homotopicspe,F1,.. .,F",...suchthat:

(i)}i~-I=Fo'"

(H)fl"((X ,A)")CB

(iii) F'"isaO-ho motopyrelative to(X,A)",-I.

Thesecanbe combinedconsecutiv elyto give a based G·homotopy

suchthatFisrelativetoAand F1(X,A)CB.•

LetXand}' bcpa t h-cc nnectedbasedG·spaces .Thena based G·map!:(X,.)--+

(1',.)isa based C·II-cquivalenccirp':(X",.)--+(yH,.)isann-eq uivalen ce fcr all

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40

closedsubgroup sJlorG (byan n-equiva lence...e meana rrrapj":(X,.)-+(l~'lsuch that/.:"'9(X ,*)-+ll'q(Y,')isanisomo rphismfor 0<q<IIandallepi morphis m forq=n).1£f:X-+Yis aba sed G·n·equi vale ncefor alln?:I, thenIis said to be abas ed G-weak hom otopy cquh·alcncc.

1£Ail isthereduced G·mappingcylinde rforfwith th eG·ma p si:X-

ii,

and r:Ai_f-+Y defined inChapte r 1, thensinceris aa·ho motopy equivalen ce,fis a G·n-cq uivalenceifandonly

ie

iis.

3.12Proposit ion.Let

J :

(X, .)-+(Y,.)bea baseda-map.Then(Ai" x,.)is G·n-co nn edcdifAndonlyifi:X^-+

u,

^{is a}^{based G·n}-e quivalence.

£.tQQ.(:Considerthelongexactsequence

.. .- +ll'll(X) ..i=-.ll'f!(Ai,)

.s;

xf(A"I"X )

..!...

:l{I(.l:)2=..,1f!~ I (A-IJ )_ ..•

wherek::;n,Uiisa basedG·n· equivalencetheni.:1l'"l~I(X)--+lI'L'_,( KI/)isan isomorphism.Thisimp liestha tkcri,

=

Im

a =

O.Sin cei.:""l'(X) _ K:'(lff, ) is anepimorphismwe havekerj.:::

ll' f!

(i f, )andhenceim

i,

=O. Byexactness 0= im

i,

= kcrd«rrf! (i l/,X).So(Ai" X )isG-n-con nected.

Conveesely,if(l\i"X) isC-n -connededthen

1rl'(M"

X):::0for0.5k::;IIand by a similarargument totheab ove,usin gexactness of thesequence,itcanbe easily shown thaii:X-+Ai,is abased G-n-e quivalcnce.•

ThekeytoprovingtheequivarlantversionoftheWhi teheadTheoremisthefol- lowinglemma.

3.13l&mIn.!!,.Let

I :

(Z,* )-+(Y,_)bea basedG-n-equivalence(IIfinite or infinite) andlet(X, A, _)beabasedrela tiveG-CW-com plexwithdim(X-A) ::;n,Theil fo r anybased G-mapsp:A-+Zandq:X- IYsucn IhatqlA

=

fop, there exist sa basedG-mapp' :X ... Zsuch thatp'IA

=

pandlor!'::::0qrela tivetoA.

f.t22{:LetAf ,bethe reducedG-mapping cylinde rof

I,

withinclu sion ma p si:Z...

Af ,

andj:Y ...

A f,

andthe G-retraclion r:M,-+Y.Provingthe lemma amoun ts toessen tiallycompletingthefollowingdiagramwherep' IA=pandlop'::::GIJ