Topological Hochschild homology of Z and Z/p

26 Pages·00.997 MB·English

by  M. Bokstedt

The topologicaHlo chschilhdo mologyo f Z and Z/p
MarcelB okstedt
Fakultatf ur Mathematik
UniversitaBti elefeld
4800B ielefeldF,R G
- 2 -
A m A
F...,..B�G L([RG(S))] BGL()R
�
m + A +
F� B LR([G(S))) � BGLR()
� G
wherteh e denotQeusi llpleunscs o nstrucBtydi eofni.n itthiteoo nts,ap la caen d
m
theb asoef+ t hel asfti braatrieco onm poneonftK sR( [ G(S) )r espectKi vRe ly)
J ( .
Thset abKl e- thiesos ryod efin,te hdaFti' s a na pproximtaota ino n
s s
m-folddel oopionfgK ( R I)np a.r tictuhlimasar k,e sK intao s pectriunam
m
canoniwcaayl.W ec omputthees tabhloem otoopfy K R([ G(S ) relattiov e
)]
KR()i nt wod ifferentF iwrnasoyttst e.h atth ef ibrathiaovnsese ctioSnisn.c e the
totsapla cient hes ecofnidb rahtaisao p nr oduscttr uctwuerh ea,v ae h omotopy
equivalence
Fotrh er elasttiavbehl oem otowpeyo btain
SincFie's m -connecttheiedsq, u atlhseg eneralhiozmeodl oogfty h es pacKeR ()
s
witcho efficiinte hnest pse ctrKu(m)R ,f osrm alil.I nt hel imoivte mr ,we o btain
equality.
s
Butt hes pectrKu(m)R i sa m odulsep ectrouvmeR r, s oi ti sa prodtu c
ofE ilenbMaecrL agn-es pectTrhahe.o molowgiytc ho effiicnit ehnistsp ectrum
isa sumo fo rdinhaormyo loggryo upwsi,tc ho efficiinte hneht osm otyog proups
s
ofK ()R.
Wec anc omuptet her elatsitvaebh loem otyo ipna diffewraeyn,nt o ticing
thasti nscet abhloem otoipsay h omology tihtde ooeryns,o tc hanugned er
thep lucso nstrucTthiiomsne .a n,ts hawte c anu set hef irfsitb ration
toc ompuit.te W eo btaai snp ectsreaqlu ence cotnovt ehrerg eilnagts itvaeb le
•
homotopIynt. h el imoivte mr, t hissp ectsreaqlu ecnocle lap,as nedws eo btain
af ormula
f A m A
(B GLR([G(S) BGL()R) H (G LR(), M( )R )
m J) = i
rr
+
Fodre taislese, , [
[ J5 11J .
Combinoiunrtg w oc alculawteig oents ,
s
H (G LR(), MR()) H (K R() ( (K)R ) )
k i i
� EB rrj
l+J=k
Inp articauslsaurm,it nhgec onjectthuarstet abKl-et heeoqryu atlosp ological
Hochschhoimlodl oagnyd, rectahlalbtiy an cgo mputaotfQi uoinl ltehne
highehro moloogfGy L (/Zp)w itcho efficiinZe /nptv sa niswheeos b,t ain
"'{
io dd
Hi(G L(/Zp), M l.(/p)) = 0Z. /p ie ven
- 3 -
Fin,a lwlIay ntto t hanIkM .ad senf ohri vse rcya rerfeudalni go fa ne arlier
versiooft nh ipas pera,n df opro intionugts eveerrarlo arnsdi naccuracies.
Ia lstoh ankW aldhausefno hre lfpuldsi cussi.o ns
F.
§1. Wea rego intgo d etermitnheeH ochscdhh oimloloogfty h er ingfungcitvoerns
byX Z[X]r espectXi vel.lp/y[ X.J
H> ��
Recafllr om2 ] itfhF a(ti- s)a c ommutartiinvfgeu nct,to hrewne
cand efinteh teo p[o logiHcoaclh scdhh oimlologTyH HF().T hiissa hypefr-s-pace
int hes ensoef[ 1J0a nd [15 J .Th imse antsh aitnp artictuhlaatr ,
ith asa r ingstruucptt ouh roem oto.p Wyec ana lsmoa kea r ingascpeo uto fF .
n n
LetF denotteh ei nfinliotoepa scpeli mO FS().T hicsa nb em adei ntao r ing
upt oh omoto,ap nydt heirsea m apF T�H HF().I npa rtictuhlesa pre,c troubmt aidn e
frotmh ei nfinliotoep struacstsuorcedit aott ehea ddtiivset ructiunTr HeH F()
a moduslpee ctrouvme rF .
is
Ifto llotwhsai tf F i sg iveans F (X=R)[XJf oarc ommutartiinv,tge h enT HHF)(
isa producto fE ilenbMaecrLga-nsep ect.Tr haaer gumeinstt,h auts intgh eu nit
map
0 0
S R[S]
�
wec anc onstrurcett raa cotfsi poenc tra
� �1\
�
IJ��i�2 I=I;IJ"ji�J IJil:Ii�J
I
Smashp roducto fa n EilenbMaecrL ga-nsep ectrwuimta hn ys pectriusm
ap roducto fE ilenbMaecrLagn-es pectrIaft.o llotwhsa,tT HH(Ri)sa r etract
ofa producto fE ilenbMaeclr agn-sepe ct.rB autth einti sa p roducto f
EllenbMaecrL ga-ne spietcste.rl af
LetK (Mn,)d enotteh eE ilenbMaecrLagn-es pecturmo fd miensino,wn h ich
corresdpst oont heR -moduleM .L et denoter estrdpi rcotducet.
IT'
co
Theor1e. ma1 ). = IT K('l. /,p 2i )
J=!;IJ"j i�L�bI
io=
co
b) = K(l. O, ) IT K(l. /i2- i1)
II:!�i�2 X I I
i1=
c) Thmea pT HH(l ,)l. THHl.(/,pl. /pi)st hep roduct
�
oft hec anonimcaaplK (,l.)O �K (l/p ,0 )w ith Btohcek smtaepisn
K(l/pi 2pi)- 1 K{l./p2p )i
I � I
d) Wec anc hootshee i somorpihnip samra t,s ot hat if
...,r:fiJ-: : \
··,;t�-H2�1< K�(l- pi ,2i l.)/ )p H2(i THH(l. /p,Z) /p)
·-. 2i I c
�
��·-\.�
•
thef undamentcalla ,st shetnh ec oprduocti nc ohomology ,
is
givebnyt hep roducti nT HH,i cso mputbeytd h ef ollowfionrgm ula
i
t = 2:t ®t
zt 2j 2i(-j)
D. jo=
�
- 4 -
� �
e)W ec anc hosteh ei somorpihnip sarbt ��hati f
,.
t _ H2-11< , Z-i1 H21-1 THH z�)'�·/ p)
2i1e K( < �
Zll ) Z,/ pc) )
ist hef undamecnltaasltsh ,et nh ec oprodiusgc vtie nb y --�·
i1-
, ( , )
t. 1= t .
!::,.2p t-1 ® 2p1-+1 2 pt1 -®1 1 �+ j ""�t = 2L p1j -® 2l p(I' -J'-) 1
Hered enottehseB ocksatsesionc itaotp e. d
�
Rema:r kPardt sm iplays sertthsat th em ultiiacptlisvter uctiusm raex imally
nontivrila.T hicso uladl sboe f ormulawtiethdho motogpryo u(pt sh er inogf
homotogpryo upissa g radpeodl ynormiina)gl P .a retc ana lsboe f ormulated
int ermosfh omotogpryo upbsu,tf otrh iosn en eedhso motogpryo uwpist fhi nite
coefficients.
Thper ooofft heore1m.w i1l olc cutphye roefts hti sse cti.Wo ena rgeo ing
toc omputthees pectrhuomm oloogfty h es pectrouftm h et opologHioccahls child
homologTyh.iw sli lb ed onbey spectsreaqlu en.c Ienos rdetroc omputthee
difefrent,ia anldts os olvcee rteaxitne nsiporno bleimnts h essep ectsreaqlu ences,
wew ilnle epdr eciinsofer mataiboonu tth heo moloogfty h e· spec.tT rhuemcsoem putoants i
will bei nd one
Thferi srt§e 3m.a trkob ed onei,st haats T HH(iRs)a p roudcto fE lienbeMragcL-ane
spect,irs tah omotopy istd yeptee rmibnyie tdhs o molo.gA yctuailtil sye ,v en
deteirenmdb yt heh omolowgiytc ho efficiineZ niftpos er a cph, t ogetwhietrh
complektneo wleodfga el hli ghBero cksmtaepi.sn
n
EacEhl ienbeMracgL -ansep ectrKu(Zm / p, m ) conttwrosi ubmumtaensd s
int hsep ecthroummo loogfty h teo poloHgoicchaslc hhoimlodl o,bg oythios moripc
,
to thed uaolf t heS teenarlogde bartpa m odultoh eB ockst.Oe nien
;4
copiyls( s� lh iftined di mensbiyom n a ndo nec opbyy m+1.T htew oc lasses
n
arree labtyet dh e hiBgohcekrs atsesionc itaotp e.d
Fixp riame p .F ronmo wo n,a lhlo mologgryo uaprsew itcho eifcifnets
� in Zp/.T hsem iplicsitarlu ctouftr oep ologHioccahls chhiolmdo lopgryo iveds
�� usw itah s pectsreaqlu ence cotnovi sets rpgeicntgrh uomm ologLyeu.ts f irst
� 1
..�..{.- '\._;.,. .c"' onsitdheecr a sTeH H(l. lp) .T heE -teromf t hiss pectsreqaule nicseg iven
/'. �-o y
..
Thfeir sdti fefrentiisga ilv ebnyt heb oundamrya pso ft hesm iplicoibajle ct.
Theisned utcheeb oundamrya psd efin(io nrgd in)aH ryo chschhiolmdo loHg (y;4
of actionng is te.l f )
;4 2
Ifto llotwhsa,Et isi somorptohH iocc hsidcl hhomoloogfy acting on
;4
its.eR lefcaf rom thafto ar c ommutartniigv Se,
11 [ 4 J
H( S ) Tor ( S , S )
� S ® S
Recaflorlm 11 that
[ 1
T'A = l/2( �1 I �2 I ' ' . ] deg 1= � z·i - 2 ( p= 2 )
= lip � Zip -
;4 [ �11 2 1 • •] ® [ ro TlI • •] I Ti- 0
1 1
deg� ·= p2- deg =p2 - 1 ( p
1 2 1 t > 2 )
T
- 5 -
TheK Unnetfho rmulaap pliteodt hec ompledxe finiTnogr saytsh aitf and
M
areb lmoduleosv ert her ingRs1 r espectivetlhye nR 1
M2 2 2�
(
TorR t® RzM1181 M2 M,1181 M2I ;;' TorR t( M1 M,1) 181 TorR z (M M2)
Let,4 'be definebdy t hef ormula
:
.A0 lJ2 [� 1® 1 ;_ 1® �11 �2® 1 -®1 � 2 I p = 2
.A'= l.lp[ � 1® 1 1-® �1 , .J.® Zip[ r '0® 1 1® - 1:0, • • • J p > 2.
Thent heK Unnetfho rmulmaa yb ea pplieidnt hep resensti tuation,
inv iewo ft het wom apsg ivebny thei nclusi;4on'� ,4 ® ,A andt hed iagonal
map ,A � ,A® ;4,re sp�ect)ivWeel oyb.t ain that
�;1 '· �
• A 7H
-�:-: �· <!;(,.4 ·c ';;,.4 ® Tor,A '( Z ip, Zip) , �
' -,)
�
Usintgh eK Unnetfho rmulaag ainw,e canf urthedre composteh eT or- factoirnt his
tensoprr oductW.e obtain
; \\1
f0 i \ '
H ( ;4;;' [,A I I • • •] I = ; degA = (1I z-1) p( 2= }
i
) A1 A2 A
H ( ,.4;;' (,.4 , , I. f,= 0 l � y ® f( y2 ). .
1
) A1, A 2 A )
i i
deg = ( 12 p,- 2 )d e;gy .= 1( , 2 p- 1 ) ( p> 2 )
A.
1 1
y.( a) ® ®
Thec lass isr epresenbtye1 d ® t. t. . . t®.· ( wheret het ensor
produchta sa +tf a1c to)r sa, n d by1 ® 1 1 1
A. �· .
Theg amma-algebrr (a 1a i s)d ef1in eads t hev ectorspoavceerZ. /pw ith
(i)
basisi venb y thes ymbolas ,a nde quippewdi tha produgcitv ebny
(i)( Ti (i (i)
. aa j = ; j ) a+ j.A n exerciisneb inomicaole fficisehnotwss t hat
0 ( )
r ( a = Z) /p [a ( p) I a ( p1 ) I • • )I a ( pi p) = 0
Thes pectrsaelq uenciess lightdliyf ferienntt h ec aseps = 2 and
p odd .I nc asep = 2 t,h em ultiplicagteinveer atorsa lalir nef iltrat1i ,os no
ford imensionraela sionasl,ld ifferentviaanlisso hn them.S incteh eP r:.9Q ·u-·c-t·i-_ ·s
compatibwliet ht hes implicfiiallt rattihoinis,m plietsh ata lld ifferentviaanliss h.
·2 ·
Thati sE,00 = E i�-thes pectrsaelq uencaes,a ringP.a ssinfgr omE 00 to
thes pectruhmo mologyw,e havea ne xtensiporno blemT.h isp robleims r esolyed
§3.
by the follloewmimnag,w hichw e areg ointgo p rovien
Lemma1 .2.L et € H* ( THH(Z /2) l.;l 2) represetnhte p ermanencty cle
Ai
c.. Ai. Then 2
)
( >::".= A
• '·-:-. i+1
1
Ju p toa no�z�rfoa_c toarn,d c ounted modulod ecomposables.
--' ""?.--�/ "... •.' ·' . ...
_,_..
Thef actt hatt heraer en o differentiinta hless pectrsaelq uencper,o ves
.,..:(\. .- -...,� ::
.. ·"(
- 6 -
thatth e� pectrum of ) Iasf remeo dule owvietreh x actly
homologyT HH(Z /2 ,4
oneg eneraltneo arce hv edne greef.o llotwhsai tnt hep roduocftE l ienberg­
It
McaLanes pectra, equivatloe nt Z/2) ,t heriese xacolnye copy
homotopy THH (
ofe acohf t hes pectKrZa(/ 2 2,)i , i0 .T haits, 11.a. follofwospr 2.
� =
11d.. follofwospr 2f rolme mm1a.2 By.c hangitnhgeh omotoepqyu ailvence
=
ofT H(HZ /2t)o t hep roducotf E lienbge-MraLcansep ecatwr,ec na arrangteh at
,n oto nlmyo duldoe coomspabloersu pt oa c onstandt .f ollows
-·(·- >�2 .'3-= . 1.1.
no1wf rodmu alaitzioInnc. sa ep iso dd, theraern eo ntraidlivf ifereanltsi.
./ 1+1
Lemma For k p-1t hedif ferendtiiasil ed ntailczleyr o, and
1.3. 1< < k
(jp) '\ ( (jp-)1 p(J-2) ��)p.,1')- 1 .
d ( )
p-1 = i+1 i i i
fj A f f • • •f
Thisw iblelp rovedin §3.
Thr·eni g 2 isi nt hisc saeg enerabtyte hdec lassAesa nd1/ pj) .S nice
E i
thec lassAesh vaef ltirianot 1 ,al dlfi feenrtiadlsf ori 1v nasiho nt hem.T he
i i >
fisrtp -1d fifeenrtialaesr t heerfoerd etmeirendb yl emma1 3..
p
We wnatt oc omputEe.
p-1
Wec na wriet theE temr asa t ensporro duct:
Ep-i ®A ®.
;;' � '
whree A A l A�®.J. (J )y . �
i =,A[ i� I r i
Thdei ffearled m apsA toi tse,·l·swfoe.c na constihdehe orm ology
�tlp-i i
9
ofA .w itrhe spetcoti t.
··-·-·-·-·--·- ·· .. .. . .
A.is t hed irescutmo ft woc opioefs ® ( ,i ndebxye da nAd.
I 1 ,A r y.) 1 + .1
Thdef iferieanmltap so neo ft hec opiteost heo theIrne. c ahd miensino conugernt
l\\\1; o \ mo·d1u l2opt her in�g (y ).h a so nec opoyf1 t hev ectoarcsel.p /.p1T he
7 t 0 .:1
dfifereandlte icraesedse grbeye1 .e claitmh atthed i(f�_arlise nntiHe .Th is,
W / ·-...-·--..--.?...:.�..···.._-· ·..·- ..--.-------'tf----�· v- .
alspor ov,eb syd imenscioounn t,it nhgatth ek enreclo nsiesxtasc tolfty h e
:!
elemenotffs i lattriloens tsha np .T oc hekct hei njeicytt,i vntohattei ts ufiefcs
•: t o!p rovteh en onvaniosfht ihnedg i fferaeoln nmt oinomailisnt hes ymbols
Tihsf ollodwriesc tlfyr otmh ef ormauf lotrh ed ifferae.ln ti
:.1
The homooflA ogw.yi trhes pecttod equals
p-1
. i
: J . 1
_i i
j (y ).P.
B. ,A=[ y.J I
�
�
I 1 1 1
�,
TheK linneftohr maus lhowtsh at
I ��
�'-.J
Thirsi nhgsa sae otf g eneraitnof risla toitrnl estsha no re qualt o It
1.
folltohwasta lhlg ihedri ffereanrtzeie arl.oAs s i nt hec saep 2t,hi ssta tement
=
- 7 -
porves1 .af 1o ro ddp .A gani,w eh avaem ulptlciiavteie xtenspiorobnl em.
Lemma Let H* T(H H(l i}p Z:;l p) r peresetnhtep ermanencytcel
.1. 4. y.E
y. Theunp,t o1a f actor,a ndm oduldoec omposables
1
=
Thpeo rofw illb eg vienin §3.
Itnh es amew aays f or p thsip orves1 .1f.o rod d dp .
= 2,
Wen owt urtno t hes pcetruTmH H()l..W e fiax p irmep .T haer gunmte
wilbled fifeentri n the tcwasoes p andp odd.
= 2
Asb efore weh avaes pectsreqaule nwciet h
I
whree H ( Zip} i st hes pectrhuomm oloogfty h eE linebegr-Malcane
;4 = * l;
spetcrum oft her ign
l.
Thsii sa s pectrlas equceeon fa lgersab over ,w hihc aerf erea s -modusl.e
;4 ,A
We fisrtt eratt hec asep
= 2._
Threni gs trutcuroef is knowns,e e Iits ap olymniaola lgerab
,4 [ 9J .
over ono neg enraetro ofd egree an_g�e nraetors ofd egree i- f1o r
'll2 1J 2, �i 2
eachi
� 2. '/_
Theri gn mapZ induces a p m Thimsa pi sg vienb y
� ;4 � ;4.
l-12
2 \
TJ I�
�1
Theirsea spectsreqaule nccoen rvgeintgot hes pcetruhmo mology
witchof efinctisie n ofT HHZ(}.U snigt her feormulatioofnH o chschil d
l12
I
homoloagsay Tora ndt heK Unneftormhu l,aw e can computteh at
I
2
(e ). 0
I =
1
Thecal sse 3i gsi vebny 1® 1Ja, ndt heca lsse2 ib y 1® i� .
00
Thcel asse.ea sl hla vfeli trtaoin 1, s oa ldlfi frenetailvsa snhia,n dE
equalE2s . 1
00
We consdierth em ulptliaitcviee xteonnssiin theE We calmi,t hawte
•
can chooserpe ersenvtaets2i i in H* (T HH()l.) o ft heca lssee2s 1 ,s ot hatth ye
�
aerr elda tbeyt hee xteonns i
Undetrh em apo fs pecrlats eqnuceeisnd ucebdyt hesm iplciailm ap
- 8-
THH(l.)_, THH(l./2)
1® - ® �· ....... ,
thec lassi r eepresenbtye d thism apst o1 -
2
�� �� ---
2
Them ap ofE terms 1
�·
\
� '
)
1® -(
sends3 reepresenbtye d 1J toz eroa,n de 2 1 toA 2 1. 't ·
We hav·ea lreadsyo lvetdh ee xtensiporno bleim" TnH H (Z./2)W.e know, �
- - -
2
thatw e canc hooscel asses representinsgo t ha(t A ) = A. Inp articular,
i i i 2i
A A .
H*( THH(l. /2) i)s a polynomiaallg ebrTah.e i mageo fH *( THH(l.)) isa subalgebra,
containitnhgei mageo f,4 a ndthei mageo fe 4. S ince4 mapst o 4 modulo
-<;_. _ e �
t"' decomposabltehsei, m ageo fe 4i sa lgebraicianJ:].dye pendeonf,At ,an d
-'� so algebraicianldleyp endoefnt th ei mageo f,A .I tf ollowtsh,a tt hei mage of
·"\ H*( THH(Z)i)nH *( THH(l./)� isa polynomailagle �rias,o morphtiocft [eJ4 .O n the
...,; �\� othehra ndt,h es quaroef e 3 ise itheerq uatlo 11 e4 orz erof ord imensional reasons.
Q
Thef irspto ssibilwiotulyd c ontradtihcatt T HH(l.)i sa producotf E ilenberg-MacLane
"'.-\
spectrTah.e rew ouldb e a nontrivki-ailn vaanrti,s incSeq !<)� wouldh aves quar�e 4.
3
�;;,, ':)\ ® 2
Itf ollow,st hatH *(T HH(l.)) contf1a inl.s/ 2[ e3 ,e 4) (I 3e > • Counting
)\
dimensiownes ,c oncludteh att hiiss i ndeeadl lo ft heh omologyI.np articular,
�-w..e..) canc hoosee is o that
2
.. �
We noteda bovet,h atT HH(Z.)i sh omotopeyq uivaletnot a producotf
EllenbeMragc-L anes pectrMao.r ep recicely,
T ITK ( i )
HH(l) = lX G I .
i
whereG aref inigtreo upsI.f w e onlya skf ora 2-primaerqyu ivalenwcee c,a n
i
assumet hatt heg roupsG .a re2 -groups.
K r
Theh omologoyf t h1e E ilenberg-MacsLpaencet rum( 'l/2, ii sa) free module
,A. ,4.
on twog enerartsoo ver TheE 00 -terma bovei sa l�srit:�em oduleo ver
4.
Ith aso neg eneratfoorr e veryd imensicoonn gruetnot modulo Counting
fr·�·��or·1.
dimensiownes ,s eet hatt hicsa no nlyb ea ccountfeodr b �duct .
We havet od etermitnhee n umber4s � 2.
i'j j i'+
We claitmh ati fi = 2 I witho dd I thenl1 . i sa tm ost2 i- A.c tualltyh,e
co
homologoyf K (l ..ll, 4i-)1 o ccurisn E ast hef reem oduleo ver,A generated
1
4i
by classeisnd imensio4nis- a1n d . Theg eneratoarrsec lasses
oft hef orma e3e4• • •e 2r+tr• espectiavee2li'y2+ , wherea ist heu niquper oduct
�1-�'t--. - 9 -
J. �,.«"....,. o:t.",·"�"
/'-
i+2
ofm ultiplicaitein;,;e ratoer2rs o fd egreze' ( j-1) .L ets be then umbero f
:g.e neratoorcsc urinignt hisp roducTth.e nt het wog eneratohrasv ef iltratsi+o'i+nl
\respectivse+lly.
'-�
Thism eanst hatt hef undamenthaolm ologcyl asisn
(K ( l.
H� 4 .t -1 !LI 1 , 4i-)1 'l.J,2 ) isi nt hei mageo ft hem ap
Let� 2r denotteh eh igheBro cksteidne,f ineidn ductivoenl tyh ek erneolf
�2r-l T.h esea re tdhief ferentiinaa l ssp ectrsaelq uencceo nvergitnogt het ensor
producotf 7ll2w itht he2 -locahlo mologmyo dulot orsion·.x- ·-ias class
Lf
inH *( ,I 2)!L,w hichi st her eductioofna clasisn H *( 7l,� , .then
�2r vanishoens thec lasxs.
The fundamentcalla siss t hei mageo fa higheBro ckstein.
( K('l ll l
Then ontrivcilaals isn H 41 i, 4i-)1 m)a pst hrou.�4.�,.. _!:. o it.
Inp articultahri,ss howst hatt heries a ne lemenitn H "'"'( C:lL S�+J- )+w 2h eret hish igher(� �...i' ;
Bocksteiisn d efineadn d nontriviTahli sr. e sulcta nb e improvebdy, n oticitnhga t
thec lasosf d imensio4nii nt heE ilenberg-MacsLpaencet ruhma sf iltratsi+o1ns, o
iti sn oti nt hei mageo f
Combinintgh estew os tatementwse ,s eet hatt heh igheBro cksteiln· i sd efined
F F �
on a nontriviealle menFot f H*( s:.Z0t.ti. I s ) . 1 ' -_1.? )
Butt heq uotient , I ist hes uspensioofna disjouinnito no f
s+1+ 1 F-Ys
smashproducotfs E ilenberMga-cLanes pectrsao, r elatitvoe t hes uspension
· of thes paceo fc omponent,si tsh omologwyi thc oefficieinntt hse 2 -primary
localizaotfi7l o ni s2 -torsioBny. i nductitohne,t orsioinnt heh omologwyi th
?..-coefficieonf tFss + i I F isa tm ost2 i'+1-torsio.Bn u tt hent,h e
2 i'+ s
higheBro cksteoipne rati�o21n• 1+ iso nlyd efinefdo rt het riviealle ment.
Itf ollowtsh atl1 l.. d iv••i d2eis'+ l ,w hichi so urc laimT.h en extc laimi,s t hat
havea ne qualit1 y= 21+ i .T his ise quivaletnott h es tatemen1t.b 1 for
2-primat.royr sioRne.c alflr om[ 2J t hatw e havea producotf
infiniltoeo ps paces
2
[l E'll/2'l.X J 2T HH( 'l..) � THH(l.)
=
LetC * C*( THH('ll) be)t hec ompledxe finisnpge ctruhmo mologyL.e t
W
* be thes tandarfdr eer esolutioofn l overt heg roupril.n[g l l2] .T hisc haincomplex
haso neg enerateori ne achd imensioans, a chani complex otvheerg roupring.
i
Them ap [l induceas m ap ofc haicno mplexe(s f orm ored etaiolns t his,
seet hed iscussioofnD yerL-ashoofp eratioonns s pectruhmo mologiyn
§2)
ll* : w * 0 c* 0 c* -?> c*
Thism ap isi nvariaunntd ert hea ctioonf !L /2.
e
Let C* bea chainr,e presentai hnogm ologcyl aswsi thc oefficieinn ts
x
- 10"'1
'l.J2r.T hati sI theries a chainy € c* I so that8 X 2r= y . We definteh eP ontryagin
J )
squar(e s ee[ 6
P ) )
: Hn( C* ·Z.,l2 r � H 2n( *C · 'Z. l2r+t
by thef ormulpa( )= lJ.*( e® ® X - 2r e1® Y ® X )•
X 0 X
Then,i fr edd enoterse ductimoond ulo2 r ,r ed(P (x)) x=2 .
2
Usingt hise xpliccihta irne presentxi,n wge obtaitnh ef ollowisntga tement
abouth omologwyi thc oefficiienn'l tls2 a ndt heh igheBro cksteoipne rations:
2 =
�2 r 1( x) x �2 r ( x ) r > 1
+
whereQ n( )y= tl * ( 1e ® Y ® Y ).
§3,
We wantt oa ppltyh its ot hec lasses.1 . In we prove
�
Q4 �3)
Lemma 1.5. ( = 0 .
M_?reo_:vb,ey t haer gnutma ebeo,lv1 � 2,s o ith ast oe qual Itf o8ll�os­
e e3B l' formulas �· Q (
atnhdat �y2 b o( u4c r) h =o oi• fce ite ,h ew oeb tnia nadbiouvceu,ts iintgvh eeC laryt afno rmulfao r e3e4),
that
2 �2 j ( -2ei ) = 0 , j � i- 2 .
thati s,� 2i-l isd efineodn e2 i. Usingt hatt heh igheBro cksteianrsed erivantsi,o
"'!e obtain t�h2ia-t1i sd efineodn a clasrse presenttihnegg enerationrd imension
. Ourc laiamb outl follow,sf inishitnhgep roofo f1 1.b .f or2 -primatroyr sion.
i j i
Thec oprodufcotr mul1a. e1..f ollowfsr omo urc omputatioofnt hem ultiplicative
structuirneH * ( THH('l))C.h ooset hei somorpn:_h iwsitht hep roducotf
Eilenb-eMragcLanes pectrsao, t hatt heg eneratoer1.sc hosena bovem apst rivially
intoa llt he-f ac)1.t oresx cepotn e.T hent 14-.i1s d ualt o �3( �4> -11,a nd �(l 41 1-. )
isd ualt o( 4e •T hef ormulfao llowosn dualizing.
Thec aseo fo ddt orsioinss imilabru,t i nvolvdeisf ferentaisaa lnse xtra
complicatiIontn h.i sc ase
• J (I- t )i2
and
them ap ,4� ,A isg ivebny �i� , � -c1·
�i 11 l.
Them ap ofs pectrsaelq uenciens ducebdy ther ingm ap � Zip isi nt his
2
caseo n theE -levealn inclusion
2
2 ® r< ® ) ® f( ® f( ) ®
f{cA 1A,2,. ..J (I A 1>• y1) .. c ,4[ 1,A ?A- ,. .]. (I .1A y0 ) y1
map
Sincteh is isi njectitvheef, i rsnto ntrivdiiaflf erenitnit ahles pectral
seqncuee
of THH(l.)i sd eterminbeyd t hef irsnto ntrivdiiaflf eenrtiailnt hes pectral