THE SEGAL CONJECTURE FOR ELEMENTARY ABELIAN p-GROUPS-II. p-ADIC COMPLETION ...

21 Pages·2001·1.51 MB·English

THE SEGAL CONJECTURE FOR ELEMENTARY ABELIAN
p-GROUPS-II. p-ADIC COMPLETION IN EQUIVARIANT
COHOMOLOGY
J. J. .P PAM dna .S .B
CARUSO, PRIDDI
G eb a etinif ,puorg-p tel GE eb a free elbitcartnoc ,ecaps-G dna tel :n eb tnairaviuqe
LET
elbats ypotomohoc .yroeht enO noisrev of the lageS erutcejnoc reads as .swollof
THEOREM. The projection GE x X-X induces na isomorphism
x;(X); GE(;Tcr+ x X),^ 2 GE(*cr x ;)xo
for all jinite WC-G complexes X.
sihT tluser seilpmi na eugolana for lareneg etinif ,spuorg tub we refer the reader ot [25]
dna yllaicepse [S] for that. eW llahs evig as tneiciffe a proof of the meroeht as tneserp
ygolonhcet smees ot ,wolla gnitrats morf the ylerup ciarbegla Ext noitaluclac [4,1.1] of
,smadA ,anedrawanuG dna relliM as a .nevig nehW ,‘)pZ(=G the meroeht si eud ot those
.srohtua ,revewoH rieht lanigiro passage morf [4,1.1] ot siht case of the meroeht devlovni
ylbaredisnoc erom Ext noitaluclac dna other work naht sruo does dna deiler no ]Sl[ for the
noitalsnart of a tnairaviuqe-non noisrev of the meroeht ot the noisrev stated. riehT paper [4]
therefore dettimo the tnemugra ni rovaf of a ecnerefer ot ,su dna we evah deworrob rieht eltit
htiw rieht .noissimrep
nosslraC [9, ]OI detelpmoc the proof of the meroeht yb gnicuder the lareneg case ot the
case G = .‘),Z( ehT krowemarf of ruo proof si the eno he ,dehsilbatse dna ti lliw eb tnerappa
that tsom of the niam saedi are eud ot .mih elihW ruo yramirp sucof si no the yratnemele
nailebA case. we edulcni a etelpmoc proof ecnis stnairav of s’nosslraC stnemugra wolla a
rebmun .snoitacifilpmisfo gnolA the ,yaw we lliw ezilareneg sih krowemarf dna sih noitcuder
of the melborp ot yratnemele nailebA spuorg-p morf ypotomohoc ot ylriaf lareneg -iuqe
tnairav .seiroeht sihT noitazilareneg si ton evissergid ecnis ruo proof of the meroeht nehw G
r,Z(= si desab no a ytilarutan tnemugra gnirapmoc ypotomohoc ot rehtona yroeht for
hcihw the suogolana tluser ylniatrec .sdloh
eW eniltuo ruo work ni $1 dna neht proceed ot llif ni the .sliated ehT stnemugra here were
sketched ni [22] dna appeared yllanigiro ni the stnirperp [12] dna [26].
$1. STATEMENTS OF RESULTS
Let kz eb a detneserper ygolomohoc yroeht no sexelpmoc-G =( .)sexelpmoc’KC-G eW
lliw evig a esicerp noitinifed ni 52. ,yllautcA we lliw yficeps a decuder yroehtib :X(& Y) no
desab sexelpmoc-G X dna Y. sihT lliw eb a ygolomohoc yroeht ni X (for dexif Y) dna a
ygolomoh yroeht ni Y (for dexif X). eW agree ot etirw
413
111 .J ,osuraC .J .P yaM dna .S .B yddirP
)1.1( ;X,(fE )-Y = ;E “( ;x Y ,)
(1.2) =)X(:E ;X_(& So) dna (fi/ Y) = ;”S(fc Y).
sA ,lausu for desabnu secaps-G X dna Y, we set
(1.3) )_Y_(@=)X(& dna ,)+Y&=)Y(G,k
where X, setoned the noinu of X dna a tniojsid dexif-G .tniopesab
eW lliw osla show how ot tcurtsnoc morf kg a detneserper yroehtib TL for each
tneitouqbus J= HIK of nehW kz=n& kr lliw eb .*Jn nI ,lareneg k: lliw dneped no the
noisnetxe K-+H-+J dna ton tsuj no the tcartsba puorg J. eW semitemos esu the noitaton
’h&K ot ezisahpme siht fact. roF H cG, we lliw evah egnahc of spuorg smsihpromosi
(e.g. ,3C ;5$ 17, III)
(1.4) +G(@ :X.\r :X($cz)Y Y) dna +G;X(@ ;X(@z)Y~A Y).
Here X ni the tsrif msihpromosi dna niY the dnoces deen ylno eb na ,ecaps-H dna G f A .X si
the desab ecaps-G detareneg yb X. ,ylticilpxE fi H acts no G _ A X aiv h(g, )K. = -hg( ,l hs), neht
,G A +G(=XH A X)/H htiw sti tnedive tfel noitca yb .G
elihW ruo smeroeht nrecnoc spuorg-p dna cida-p ,noitelpmoc r)X(@ si ton etiuq the
thgir gniht ot .yduts Rather, we enifed
(1.5) ;X(Q Y) = ;,X(&(mil Y),^),
where X, snur revo the etinif sexelpmocbus of X. ehT( reader yam prefer ot kniht ni smret of
).spuorg-orp esehT spuorg emoc htiw noisnepsus smsihpromosi ni htob .selbairav nI order ot
evah gnol exact secneuqes detaicossa ot snoitarbifoc ni htob ,selbairav we agree ot tcirtser
morf won no ot sexelpmoc-G Y htiw etinif ateleks dna ot emussa the gniwollof etinif epyt
.sisehtopyh
)6.1( Each puorg z$( Y) si yletinif detareneg fi Y has etinif .ateleks
sihT noitpmussa has the gniwollof .ecneuqesnoc
AMMEL 1.7. For any subquotient J of G. each group @(X; Y) isfinitely generated if X is a
finite xelpmoc-J and Y is n xelpmoc-J Jvithjinite skeleta.
roF J c G, siht swollof morf (1.6) dna htob parts of (1.4) yb na ysae .noitcudni ehT lareneg
esac lliw wollof morf (3.2) .woleb
esU of esrevni stimil ni (1.5) setutitsbus for the wedge ,moixa tub we eton yllacitehtnerap
that the spuorg ;X(@ Y) dna ;X(@ Y),^ are yllausu .cihpromosi
AMMEL 1.8. If X and Y etah finite skeleta and ,O=)Y;”X(@’mil then
Q(X;Y)$ .)~)Y;”X(@(mil;
ehT ysae proof si nevig ni $2. ehT noisulcnoc seilppa ot 52(X).
eW hsiw ot enimreted nehw the larutan pam
+GE&+-)X(;T;I )XA
si na .msihpromosi Let GE eb the rebifoc of the noitcejorp EG, -+S’, that ,si the decudernu
noisnepsus of EG htiw eno of sti enoc stniop as ,tniopesab dna eton that .’S=‘)GE( tI si
tnelaviuqe ot enimreted nehw GE(& A .0=)X sA a tnairaviuqe-non space, GE A X si
.elbitcartnoc dna ti si larutan ot ask nehw 0=)X(@ for lla elbitcartnoc secaps-G X. yB the
SEGAL CONJECTURE FOR ELEMENTARY ABELIAN p-GROUPS--II. 415
gniwollof ,ammel whose proof si osla nevig ni $2, siht noitseuq si on erom lareneg naht the
lanigiro .eno
AMMEL 1.9. Assume that i?;” canishes on contractible H-spacesfor all proper subgroups H
of .G fI )X$6 =Ofor any one contractible G-space X such that Xc = So, nellc @ rnnishes on
contractible G-spaces.
gniwolloF ,nosslraC we esu the gnirebifoc EG, -+S’+EG ni the dnoces elbairav ot
niatbo the latnemadnuf exact ecneuqes
(1.10) . . . -@(X; EG+)-~~(X)-,~~(X;EG)q~~4,C1(X; EC+)-.
ehT spuorg ;X($! EG,) yrrac the free part of the ;melborp the ;X&/ )GE yrrac the ralugnis
part. eW llahs evorp the gniwollof tluser tuoba the ralugnis part ni .43@ eW evah ylbissop
tnereffid tnairaviuqe-non seiroeht kE,H detaicossa ot spuorgbus H of G. dna we tel *j_ = ;Til G
dna *il = .,:k
THEOREM .A Suppose that kf wnishes on contractible J-spacesfor all proper subquotients J
.G‘fo Let X be a G-spuce such that XG = So and XH is contractible for all proper subgroups H.
)i( fI ’G is not elementary Abelian, then @(X; EG) = 0.
)ii( ,‘),Z(=Gff then l&X; EG) is the direct sum @rpfo 2,I‘ copies ofIYr- ‘j*(S’).
pU ot ypotomoh-G ,epyt there si ylno eno X as deificeps ni the meroeht (e.g. [ 13]), dna we
llahs yalpsid na ticilpxe ledom ni $8. yB the meroeht dna ,)Ol.l( nehw G si ton yratnemele
nailebA we nac ylno evah &(X) = 0 fi ;X($/ EG,) = 0 dna nehw G si yratnemele nailebA we
nac ylno evah 0=)X(:l fi the gnitcennoc msihpromomoh d ni (1.10) si na .msihpromosi
oT yduts ;X(@ ,),GE we deen eno sisehtopyh ot wolla noitcuder ot a tnairaviuqe-non
melborp dna rehtona ot erusne ecnegrevnoc of the tnaveler smadA lartceps .secneuqes htoB
are ylraelc deifsitas yb .~CT eW yas that k: si tilps fi there si a larutan pam :C k*( W)+kT;( W) for
spaces W regarded as laivirt-G secaps-G hcus that the etisopmoc
k*( W)ikr;( W)zk;(G x W)%k*( W)
si na ,msihpromosi where :CT G x W+ W si the .noitcejorp sihT noitidnoc seilpmi
smsihpromosi (e.g. [25, ;29 3, ;5$ 17, )]@II
(1.11) ;X(& )Z r ;G/X(*R )Z dna &( ;’tC Y) r z*( ;W ,)G/Y
where X dna Y are eerf-G yawa( morf rieht esab )stniop dna v6 dna Z are .laivirt-G eW yas
that k, si dednuob woleb fi Eq(So) = 0 for lla yltneiciffus llams .I( eW llahs evorp the gniwollof
tluser ni $8, gnisu a ecnegrevnoc tluser for esrevni stimil of smadA lartceps secneuqes nevorp
ni 57. Let k eb the murtceps gnitneserper k*.
THEOREM .B Let kg be split and k, be bounded below. Let X be CI G-space such that
XG = So and X” is contractible for all proper subgroups H.
)i( fI G is not elementary Abelian, then !&X; ),GE .O=
)ii( fI G =(ZJ and O=)k(qH f or all suficiently large q, then @JX; ),GE is the suw~ of
P -‘(‘ 2ij‘ copies of Z%*(SO).
nA etaidemmi noitcudni morf (1.9) dna smeroehT A dna B sevig the gniv\ollof
noitazilareneg of s’nosslraC noitcuder .meroeht
614 J. Caruso. J. P. !aM dna .S B. Priddq
THEOREM C. Ler G be afinite p-group bvhich is not eletwnrary .-lbelian. Let eb-gk a theory
such that
)i( :k wnishes on contracrible J-spaces for i/lc elettxrzrur~ Aheliotz subquorients J oj‘G;
)ii( ;k s1 split atd (k, *)il is bounded below for a11 non-eletnenrary .4belian subqrtorietm
J = K‘,H of .G
Then l?; ranishes otl contrnctible J-spaces for (I// subquorients J ,G’fo including G :flesri
suhT we etartnecnoc drawrofecneh no the case G .l),Z(= nehW )ii( of smeroehT A dna B
htob .dloh we nac ylno expect ot evah )X(@ = 0 )’S(“;jfi = P(SO). sihT shows the ytissecen of
the evitcirtser lacigolomohoc sisehtopyh ni )ii( of meroehT B ecnis there are ytnelp of
seiroeht kz for hcihw 0=)X(@ tub ,)’S(P#)’S(“;i for elpmaxe tnairaviuqe yroeht-K dna
tnairaviuqe ypotomohoc htiw stneiciffeoc ni tnairaviuqe gniyfissalc spaces. nI ynam cases,
tcerid noitaluclac of ;X(:6 EC,) smees ylevitibihorp ,tluciffid dna noitanimreted of these
spuorg sllaf tuo as na noitacilpmi of a tnereffid proof that .0=)X(@ See [22] dna [24] for
suoirav selpmaxe dna .selpmaxe-retnuoc
nI weiv of (1.9) dna ,)Ol.l( the gniwollof tluser won setelpmoc the proof of sht lageS
.erutcejnoc
THEOREM D. Let G = (Z,)’ clnd ler X be a G-space such rhat Xc = So and XH is cotmxcrible
fi)r all proper subgroups H. Assume that the Segal conjecture holds Jbr G ’),Z(= s‘fi < r. Then
d: ?&(X; l?G)-irY,- ‘(X; EG _)
is atI isotnorphistnfor all q.
ehT ytilibissop of hcus a proof was detseggus ni s’nosslraC tnirperp [9]. ruO tnemugra
was deniltuo ni .]ZZ[ s’nosslraC dehsilbup paper sniatnoc a retal sketch of a erom
lanoitaluclac tnemugra gnola ylhguor the emas senil as sruo ,ol[ .ppA B].
yB )ii( of smeroehT A dna ,B 6 ni meroehT D si a msihprom of free .seludom-)0x(*5 suhT ti
seciffus ot show that 6 si a noitcejib no ,srotareneg that ,si that 6 si na msihpromosi nehw
q=r- 1. nI siht degree, 6 si a msihprom of free seludom-,2 no the emas rebmun of
.srotareneg tI 1lin suht eb na msihpromosi fi ti semoceb na msihpromosi nehw decuder
dom p, dna siht lliw dloh fi ti semoceb a msihpromonom nehv\ decuder dom p.
eW esu a ytilarutan .tnemugra nI $5. we yalpsid a etiuq elpmis yroeht :k htivi tinu
:yt $/+-;TcT hcus that $! sehsinav no elbitcartnoc J-spaces for lla stneitouqbus J of ,G
gnidulcni G .flesti ecniS the ecneuqes (1.10) si larutan ni ,seiroeht we niatbo a evitatummoc
margaid
-;?i ;X(‘ ;X(,’ri-L)GE EG_)
(1.11) I‘ lr
I i
:X(&~)G~;X(‘& .),GE
ecniS )X(@ = 0, the mottob pam 6 si na .msihpromosi tI therefore seciffus ot show that the tfel
lacitrev arrow q semoceb a msihpromonom nehw decuder dom p. e’t\ llahs yfirev siht ni $6
dna so etelpmoc the proof of meroehT D.
$2. PRELIXlIS.iRY DEFlSITlONS ASD LEXIM;\S
eW ekam ruo snoitinifed esicerp dna evorp (1.8) dna (1.9) here.
SEGAL CONJECTURE FOR ELEMESTARY ABELI.AN p-GROCPS-II. 417
Let ;C eb the mus of ylbatnuoc ynam seipoc of each of a set of sevitatneserper for the
elbicuderri laer snoitatneserper of .G eW emussa nevig na renni tcudorp no .U dna we enifed
na gnixedni space ni‘[ U ot eb a etinif lanoisnemid bus renni-G tcudorp space. fI Vc :tI we
tel 2-w eb the lanogohtro tnemelpmoc of V ni ..tI A set of gnixedni spaces & si lanif-oc fi
sti noinu si lla of 6. roF ,ssenetinifed the reader yam tnaw ot take c’ ot eb the mus of
ylbatnuoc ynam seipoc of the laer raluger noitatneserper Reg dna ._J ot eb the ecneuqes
n( Reg .)O>n!
A murtcepserp-G k, dexedni no a lanif-oc set .zf stsisnoc of desab G secaps- k, /I for V ni
.d dna desab spam-G :G :E ,k“-”I Vb,k+-V for Vc /tC ni d. Here X‘,Z X= A S’. where VS si
the tniop-l noitacifitcapmoc of V. eW eriuqer G ot eb the ytitnedi fi =’b dnaw ot yfsitas the
tnedive ytivitisnart noitaler for Vc CW 2. oT diova lacinhcet ,smelborp we emussa that k,
si a murtcepserpVCC-G ni the esnes of [17, .]@I sA denialpxe there, siht noitpmussa stluser
ni on ssol of .ytilareneg ehT( noitcnitsid neewteb artcepserp-G dna artceps-G si osla
denialpxe ni ;]71[ we t’now esu artceps-G here.)
Now tel J= H/K, where K Q H c G. evresbO that the dexif tniop space ’U sniatnoc lla
elbicuderri snoitatneserper of J yletinifni .netfo eW yam tcirtser ot a lanif-oc set d ni U
hcus that KV = W’ seilpmi =V W for V dna ni’B d. roF ,elpmaxe zt{ cjgeR evoba seifsitas
siht noitidnoc for yna K cG. roF a murtcepserp-G k, dexedni no ,& we niatbo a J-
murtcepserp ,il dexedni no the gnixedni set =’d { 1”I ,:&.E;T ni ”U yb gnittel
k,( )KV ,k(= ;“)V the larutcurts spam-J of kJ are deniatbo yb passage ot dexif-K stniop morf
the larutcurts spam-G of k,. nehW K =e, k, si tsuj k, regarded as na .murtceps-H
A ecaps-G X has a noisnepsus murtcepserp-G XxX htiw Vth space E’X. ehT detaicossa
murtcepserp-J si KXzC fi J= H/K. nI ,ralucitrap fi k, si the sphere murtcepserp-G
,oS.rC=6r7 neht k, si the sphere murtcepserp-J 7r,. nI ssel yratnemele cases, the -itnedi
noitacif of k, si ssel .suoivbo roF ,elpmaxe nehw k, stneserper yroeht-K or ,msidroboc ti
does ton wollof that k, osla stneserper yroeht-K or .msidroboc See [17, 11$9] for rehtruf
.noissucsid
eW yam yfitnedi GU htiw R” dna so debme Rq ni U for lla 4 20. roF a etinif desab
xelpmoc-G X dna yna desab xelpmoc-G Y, enifed
;X(tE 4R-YC[miloc=)Y X AY k,V],
zV Rq
~l)~G;FAY(’R-V~miloc,X[~ fi q90
1’3 R’
dna ;X(q& Y) = ;XqE(@ Y) fi q > 0. ehT dnoces mrof of the noitinifed osla seilppa ot etinifni
,sexelpmoc-G tub ti lliw eb laitnesse ot ruo work ot kniht ni smret of the tsrif .mrof Here, fi
,O=q the timiloc si nekat htiw respect ot the setisopmoc
[P’X, AY k,V],- ““-” [C”‘X, Yr,Fk,V]~(‘[PX, Yr\ k,W],,
dna ylralimis for other seulav of q. erehT are noisnepsus smsihpromosi dna exact secneuqes
detaicossa ot sgnirebifoc ni htob .selbairav roF etinifni lanoisnemid X, there si osla a ’mil
exact ecneuqes for the noitaluclac of ;X(@ Y) ni smret of the ;nX(@ Y), dna the gniwollof
ciarbegla noitavresbo no the noitatummoc of esrevni stimil htiw cida-p noitelpmoc
seilpmi (1.8).
AMMEL 2.1. Let },A{ be an inverse sequence ofjnitely generated Abelian groups such that
,A’mil = 0. Then the natural homomorphism
&Amil( +- ;),A((mil )
is an isomorphism.
814 .J .osuraC J. .P RI! dna .S 8. yddirP
Proof: roF a dexif I( > 0, redisnoc the two xis mret ’miI exact secneuqes deniatbo morf the
two short exact secneuqes
,l-.qp+-,4.+-,4.,+0 -0 dna 0-+pqA,-+,4,+.-l~, p~A,-+O.
Here ,A, si the lenrek of :4p ,&+--,A dna si suht .etinif htoB ,4.,’mil 0= dna ’mE ,O=,Aqp the
rettal esuaceb ti si a tneitouq of ,4.lmil .O= eW edulcnoc that the deyalpsid secneuqes niamer
exact no passage ot esrevni .stimil eroferehT
,4,mil ),4,‘”p!,A(mil=,l;mil’p
for each 4. ehT noisulcnoc swollof yb egnahcretni of stimii revo q dna .II
eW edulcnoc siht noitces htiw the gniwollof desimorp proof.
.)9.l(~o~~~~P ecniS ,oS=GX we evah a rebifoc ecneuqes .oS/X+-X+-oS X’ eb yna
other elbitcartnoc .ecaps-G gnikaT hsams ,stcudorp we niatbo a rebifoc ecneuqes
S+-’X A ’X-X A (X/SO).
tI seciffus ot prose that &X’ A 0=)X dna ’X$@ A .O=))”S/S( eW mialc tsrif that
EE( AW 0=)X for yna xelpmoc-G .VI ecniS the zero noteleks v dna the lateleks
stneitouqbus w”/ -’Vh ’ for II > 0 are wedges of secaps-G of the mrof H/G + A S” dna ecnis
ew yam as llew emussa that si‘+J ,etinif we dnif yb noitcudni revo ateleks dna esu of -nepsus
nois that we deen ylno yfirev the mialc for .,)H/G(=’H fI =H ,G siht sdloh yb .sisehtopyh
fI Hf ,G neht +)H/G((z& A ,)X(;Eg)X hcihw si zero yb the noitcudni .sisehtopyh eW
mialc txen that ’X(@ A O=)Z for yna xelpmoc-G 2, hcus as ,’S‘:X hcus that GZ si a .tniop
gniugrA as ,evoba we deen ylno yfirev siht nehw Z ,)H!,G(= for a proper puorgbus H,
dna here niaga the noisulcnoc sdloh yb the noitcudni .sisehtopyh
93.
WBQUOTIEKT THEORKS. FA;LIILIES, ASD S-FUNCTORS
sihT noitces sevig lareves seiranimilerp dedeen for the proof of meroehT ,A tub we nigeb
yb gniyas a tib erom tuoba tneitouqbus .seiroeht ,esruocfO the seiroehtib kf for stneitouqbus
J are denifed the emas yaw as the seiroehtib kz. ,revewoH there si na gnitanimuIli evitanretla
,noitpircsed eud ot ,elbonetsnoC hcihw sekam (1.7) .raelc tI si desab no na yratnemele
noitcurtsbo citeroeht noitavresbo for hcihw we llahs evah rehtruf esu .yltrohs
A ylimaf g ni G si a set of spuorgbus desolc rednu .ycagujnocbus roF a ylimaf 3 dna
xeipmoc-G X, we tel -,X eb the xelpmocbus gnitsisnoc of those stniop of X whose
yportosi spuorg are ton ni .P’. erehT si a lasrevinu“ ”ecaps-9 -,E deziretcarahc pu ot
ypotomoh yb (E9)H = 0 fi y.fhH dna (EstH 5 >rp{ fi .F.EW Let F.E eb the rebifoc of the
noitcejorp +FE ,OS“- that si the decudernu noisnepsus of Ed. nehT oS=H)F.E( fi F$H
dna H)FV8( v‘ >tp{ fi HER. ,revoeroM the ypotomoh-G epyt of e.9 si deziretcarahc yb these
seitreporp [13]. roF ,elpmaxe X ni smeroehT ,A B dna D si P.E where B si the ylimaf of
proper spuorgbus of .G
htoB E9 dna E.9 nac eb nekat as sexelpmoc-G htiw etinif ,ateleks dna we shah ylno
deen ot ylppa (1.6) ot sexelpmoc-G Y of siht .mrof
LEMMA 3.1. For G-complexes X and Y, the inclusions X,s--+X and PE+’S induce
bijections
[X, i?F A Y&--&, Es A YJc+-[X,, .GIY
SEGAL CONJECTURE FOR ELEMESTARY ABELIAN p-GROUPS--II. 119
PROPOSITIOX 3.2. Let J = ,K/H rvhere K < H c ,G and ler .9[K] be the family of sub-
yroups of H which do nor conrain .K Then, fi)r finite J-complexes X and arbitrary
J-complexes Y,
i;:(X: Y) 2 E;(X, E9 ]K[ A Y).
Proof: roF sexelpmoc-H W dna ,Z W,F[KI = WK dna
.,]zA]K[sE,w[r,]z,~~~[=J]~z,~w[
ehT noisulcnoc swollof nopu gnittel nur-E hguorht X’R-VC (or Xy-CyC fi y )O< as Z snur
hguorht AY k,V.
ehT gnitrats tniop of the proof of meroehT A si the case 9 = }e( of (3.1), hcihw reads as
.swollof Let XS etoned the ralugnis set of a ecaps-G X, yleman the ecapsbus of stniop htiw
laivirt-non yportosi .puorgbus
LEMSI.-\3.3. For G-complexes X and Y, the inclusions SX-+X and So-E, induce narural
bijecrions
[X, EGA Y-j, + [SX, EGA Y],+[SX, Y-j,.
,eroferehT for etinif sexelpmoc-G X,
:I(S[miloc Y-R4X),kGV]G fi qk0
L.2
(3.4) ,X(@ )GE = R’
,)X9-EYE(S[miloc k, V], fi .O<q
k
eW nac ecalper the rotcnuf S no the thgir edis of (3.4) yb other elbatius ,srotcnuf dna evi llahs
evorp meroehT A yb gnitamixorppa S yb na tnelaviuqe deretlif rotcnuf htiw ylticilpxe
elbaluclac .stneitouqbus
nosslraC deifidoc the etisiuqer snoitidnoc no srotcnuf ni sih noiton of na ”rotcnuf-S“
,ol[ .]VI ,ylfeirB na rotcnuf-S (T, T) si a rotcnuf T morf the yrogetac of desab sexelpmoc-G
ot flesti together htiw a larutan pam
7: T(X A Y)+(TX) A Y
hcus that 7 si the ytitnedi fi =Y ,oS T seifsitas the tnedive ytivitisnart noitaler no T(X A Y A ,)Z
dna 7 si a msihpromoemoh nehw G acts yllaivirt no Y. ,yllautcA( we lliw ylno ylppa T ot etinif
).sexelpmoc roF the ralugnis set rotcnuf ,S the pam S(X )XS(+)Y A Y si tsuj the .noisulcni
A
roF yna rotcnuf-S T, we enifed ;X(@ T) yb gnicalper S yb Ton the thgir edis of (3.4). roF
.O=g the timiloc si nekat htiw respect ot the metsys of setisopmoc
’-”I [Zw-”
-GlvGk,)X':~(TC T(Z’X): C”-‘kc V],=[T(EwX), k, WIG
for Vc IV, dna ylralimis for other seulav of q. ehT spuorg ;X&i T) are larutan ni X.
fl T sevreserp rebifoc ,secneuqes yeht evig the smret of a ygolomohoc yroeht no etinif
,sexelpmoc-G tub we t’now deen siht fact. nI enil htiw the noitinifed ;X$/fo Y) ni (1.Q we
enifed
(3.5) ;X($/ T) = ;,X(@,mil r),^
no etinifni sexelpmoc-G X, where X, snur revo the etinif .sexelpmoc-bus oT erusne that mil
sevreserp exact secneuqes here, we eriuqer ;X(@ T) ot eb of etinif epyt nehw X si .etinif roF
the srotcnuf-S we llahs ,esu siht lliw wollof ylisae morf (1.7) dna the lanoitaluclac pihsnoitaler
of the seiroeht ;?(@ T) ot the seiroeht ;k
024 J. .osuraC J. .P yaM dna .S .B yddirP
A pam :4 ,T( (+-)r ,’T 7’) of srotcnuf-S si a larutan noitamrofsnart :# T-t T’ hcus that
~$’r Q(= A 1) :r T(X A Y)-Y- +(T’X) A
for lla X dna Y. eW yas that 4 si na ecnelaviuqe or a noitarbifoc fi each tnenopmoc pam
:4 TX-T’X si a ypotomoh-G ecnelaviuqe or a .noitarbifoc-G eW dnetxe the lausu
snoitcurtsnoc of ypotomoh yroeht ot srotcnuf-S .esiwecaps suhT wedges. hsams stcudorp
htiw spaces hcus( as senoc dna ,)snoisnepsus ,stuohsup ,srebifoc dna so no lla tsixe ni the
yrogetac of .srotcnuf-S fI :4 T-T’ si a ,noitarbifoc we niatbo a tneitouq rotcnuf-S T’/T
htivL (T’jT)(X)= T’X/TX dna a lacinonac ecnelaviuqe of srotcnuf-S C,-T’iT.
A pam :4 T+ T’ secudni a pam :*+ ;X(@ T’)-+LE(X; I). fI 4 si na ,ecnelaviuqe neht 4*
si na .msihpromosi if c$ si ,noitarbifoac neht $* stif otni a gnol exact ecneuqes
. . . + c&X; T’/T)-&(X; T’) 5 ,@(X; T)-&+‘(X; +)T';'T . . .
ehT proof sesu the fact, deilpmi yb the msihpromoemoh noitidnoc ni the noitinifed of na
,rotcnuf-S that
ylraelC ;X& T v T’)z &(X; T) 0 &(X; T’) for yna S srotcnuf T dna T’.
ehT stneitouqbus of ruo deretlif noitamixorppa of S lliw eb wedges of snoisnepsus of
srotcnuf-S of the gniwollof lareneg .mrof
NOITINIFED 3.6. Sqqxxe ueuig spuorgbus CK H c G. Define an S-functor .K(C H) b_v
ietting
,K(C H)(X) = G _ A .KXH
ehT pam-G :T ,G A H(XR A _G(+)KY A HX’) A Y si the noisnetxe oat pam-G of the tnedive
noisulcni of secaps-H KX A +G(+KY A )KXH A Y.
tneitouqbuS seiroeht retne otni ruo work esuaceb of the gniwollof .noitavresbo
AMMEL 3.7. For HaK c ,G ;X&! ,K(C H)) is isomorphic to l&s(X”;).
Proof. tI seciffus ot evorp siht for etinif X, erofeb passage ot cida-p .noitelpmoc roF
lanoitaton ,yticilpmis we ylno evig the noitacifirev ni degree 0. Here the snoitinifed dna
suoivbo smsihpromosi evig
;X(@ ,K(C H)) = G[miloc + ,6X‘“ k, ~J’J~
A HC
2 ,KX“’LZ[miloc k, qH
2 Z[miloc ,KX’v (kG Y)R]H,‘K= 5; s(XK).
.Y> NA N.0ITAMIXORPPA
OF THE SISGULAR SET FUSCTOR
ruO noitamixorppa of S si a tnairav of s’nosslraC ,ol[ V]. We show the tahwemos
gnisirprus fact that, pu ot ypotomoh-G ,epyt XS nac eb detcurtsnocer yllairotcnuf morf sti
dexif tniop sets XH for yratnemele nailebA spuorgbus H. nosslraC( sesu lla proper
spuorgbus ;ereh siht eciohc secudortni etiuq a tib of extra work, hcus as ,ol[ VI 7-6 dna
V ).13-l ,ylesicerP we llahs evorp the gniwollof .tluser
SEG.\L COSJECTURE FOR ELEMENTARY ABELIAN .II--SPUORG-p 171-
THEOREM 4.1. Let G be a finite puorg-p of p-rank r. There is an S-finctor A and an
equicalence $: .S-A A has a noitartlif
cA,FcA,,F . . A=A,_,Fc
by successire cofibrations. If ,B = A,,F and ,B = ,F/A,F _ A, fbr 0 <q < r, there are iso-
smsihprom of srotcniij-S
,B 2 ,)o(A(CqCV H(w)).
lm[
Here the o are strictly ascending chains ,,,A( . . . , )qA of non-trivial elementary Abelian
subgroups of G, )w(A = ,,A and
iA=‘-g,Ag/g{=)w(H for .)qbi<O
The wedge runs over one o in each orbit [o] under the conjugation action of G on the set of
such ascending chains. If G = (ZJ, there is also an S-fitnctor A bcith a filtration
&F&F . . A=A,_,Fc
by successive cojibrations. If &.,F=,& dna =,B _,F/A,F A, for ,1-r<q<O there are
isomorphisms of S-functors
B, 2 /,/FJc( ,)G,)u(A
0,
erehvl the wedge runs ocer the strictly ascending chains ,,,A( . , )qA of non-trivial proper
subgroups of G and )w(A = .,A Moreocer, there is a cofbration A+-A such that the quotient
A/A is equivalent to the wedge of 2”1-r(rp copies of the S-jimctor I’-‘C(G, G) which sends X
to xr-lXG.
Before gnivorp ,siht we show how ti seilpmi meroehT .A llaceR that Y&=X ;ereht that ,si
GX = oS dna XH u {pt> for H # G.
Proof of(i) of Theorem A. yB (3.4) dna the ecnelaviuqe A-+&
E~(x;EG)aE~(X;S)zE~(x;A).
yB (3.7), the tneitouqbus srotcnuf-S ,B of A yfsitas
;X(,“;I B4) z ;X(“-;;&I C(A(o), H(o))) z &k&$XA’“‘),
IO1 0
where J(o) = H(o)/A(o). fI A(w) # G, neht X ‘(N si a elbitcartnoc J(o)-space ,dna ecnis
A(o) # e, IIJ*(o)(XA(U))=0 yb ruo sisehtopyh that ET sehsinav no elbitcartnoc J-spaces for lla
proper stneitouqbus J. fI G si ton yratnemele ,nailebA neht A(w) t’nac eb G dna Ez(X; BJ = 0.
eroferehT ;X(@ A)=0 yb noitcudni pu the .noitartlif
Proof of (ii) of Theorem A. Let G = ’),Z( dna llacer that *j = k&,. fI r = 1, neht .B = A dna
;X(@ B,) )’S(L?jr ecnis )G(=o si the ylno elbissop .niahc sihT sevorp the tluser ni siht case, so
emussa that r 2 2. yB the proof of part ,)i( ;X&! )A = 0. eroferehT
ecniS !$(X; C(G, ,)’S(*^j=))G ;X(@ )G?, si the mus of ~‘(‘-l)‘~ seipoc of .)”S(p‘-Y
ehT proof of (4.1) si desab no saedi of nelliuQ [28,29]. Let d )G(’i&= eb the poset of -non
laivirt yratnemele nailebA spuorgbus of .G eW regard .&’ as a yrogetac ni the etisoppo of the
lausu ,noihsaf gnidrager na noisulcni A c B as a pam .A+-B ehT puorg G acts no d yb
114 J. .osuraC J. .P yaM dna .S .B qddirP
noitagujnoc of .spuorgbus fI G ,r)pZ(= we tel ?s. c.d eb the poset of laivirt-non proper
spuorgbus of .G Here G acts yllaivirt no .d dna .g ecnis G si .nailebA
Let 5 etoned the gniyfissalc space rotcnuf no )lacigolopot( seirogetac [32]. tI si deniatbo
yb gnivlppa cirtemoeg noitazilaer (e.g. [20, $111) ot ,sevren where the evren of a yrogetac si
the lausu laicilpmis space htiw secilpmis-q the selput-q of elbasopmoc ;sworra see [32]
or .]l$.S2[ ehT rotcnuf B seirrac seirogetac-G ot ,secaps-G dna the gniwollof si a thgils
noitarobale of [29, 2.21.
AMMEL 4.2. IfG fe, then d.B si G-contractible. In particular, (B&)H is non-empty and
contractible for erer_v H c .G
Proof Choose a lartnec puorgbus B of order p. ecniS A c ,=BA B dna ~.EBA for ,NOEA
we evah larutan snoitamrofsnart morf the rotcnuf BAHA ot the ytitnedi rotcnuf of .,d dna
ot the tnatsnoc rotcnuf at .B esehT srotcnuf dna snoitamrofsnart are tnairaviuqe yb the
ytilartnec of ,B ecneh yeht ecudni seipotomoh-G gnitcennoc the ytitnedi pam of &.B ot the
tnatsnoc pam at the xetrev B no passage ot gniyfissalc spaces.
eW tcurtsnoc ruo noitamixorppa of the ralugnis set rotcnuf yb gnizirtemarap d yb
ralugnis stniop of secaps-G X. eW agree ot regard X as a yrogetac-G htiw tcejbo dna
msihprom spaces X dna htiw larutcurts spam ,ytitnedi( ,ecruos target, )noitisopmoc the
ytitnedi pam of X. Of ,esruoc the gniyfissalc space of siht yrogetac-G si tsuj X kcab .niaga
eW weiv XS .ylralimis
DEFHTION 4.3. Define a topological G-category J[X] and a continu ous fitnctor +k
sZ[X] -SX a.sfol/o\cs. The objects of &[X] are pairs ,A( x), where A E .&’ and x E X”. There is
a morphism ,A( ,B(-)x )y whenever B c A and y = x. The group G acts on objects by ,A(g x)
,i-gAg(= gx). The set of objects is topologized as the disjoint union of the spaces X\. The set
of morphisms is topologized as the disjoint union otter pairs B c A of the spaces XA. Thefunctor
$ is given by the X-coordinate of objects and morphisms.
PROPOSITIO4N.4 . roF yna X, XS+]X[&.B:$B si a G-homotopy equivalence.
Proof: yB the daehetihW-G ,meroeht ti seciffus ot evorp that H)XS(+-H)]X[d.B(:H)/CrB(
si a ypotomoh ecnelaviuqe for each Hc .G eW nac pass ot dexif stniop no the level of
seirogetac dna ,srotcnuf dna the gniyfissalc space rotcnuf setummoc htiw passage ot dexif
.stniop yB s’nelliuQ meroeht A [28] hcihw( sdloh for lacigolopot seirogetac ecnis the niam
tupni si the lacigolopot fact that cirtemoeg noitazilaer seirrac esiwecaps secnelaviuqe ot
secnelaviuqe [21. ,)]4.A ti seciffus ot evorp that /“$(B x ) si elbitcartnoc for each .“)XS(ex
ehT ammoc yrogetac x./“$ has stcejbo ,A( x) htiw XEX” dna A dexif yb H dna smsihprom
,A( ,B(-)x x) htiw B c ,A ,~XEu. dna A dna B dexif yb H. fI ,G si the yportosi puorg of ,y.
neht H c G, dna x/“$ si tsuj a ypoc of .‘#G(&. suhT (4.2) sevig the .noisulcnoc
eW are detseretni ni desab sexelpmoc-G X. ehT noisulcni of the tniopesab * ni X secudni
a noitarbifoc-G .]X[&B-]*[&S ylraelC $B factors hguorht the tneitouq pam
,]*[~~Bi,]X[~.B-]X[~B dna siht tneitouq pam si a ypotomoh-G ecnelaviuqe ecnis
]*C&.B si .elbitcartnoc-G enifeD
AX=&d[X]/B-Qu’[*]
dna tel 4 etoned the decudni ypotomoh-G ecnelaviuqe .XS+XA neviG a dnoces -G
xelpmoc Y, enifed the yrogetac-G Ld[X] A Y ni the tnedive yaw dna enifed a rotcnuf-G
..d[X A ]X[d+-]Y A Y