KMCX [ Xp I ( 1km ) Ku ) = spectrum , , generalized - PDF document

- homology & K spins twisted Twisted bordisue - ( ) On khoraui Baum & Schick with work joint , §1 Geometry homology untwisted & K i - K ( X ) defined Vector geometrically via - , bundle X over space . theory 4 generalized cohomology KMCX [ Xp I ( 1km ) Ku ) = spectrum , , generalized theory homology → Skt [ " Ku(X7 ] Ku X eoliin = a + , k a →

- theory Poincare for K duality [ MINE or ) M Ku ( spins closed → . Es * ( M ) M ) Ku ( K * . [ M ] tn x nx . Bolt M 24 by periodicity Or di = - , , Koch ) Howe ( 2 Koch 1 → ) , ( Bmok ) ) m ] C [ E - vbl in and

* tiyah set oh equivalence 's Koen idea , : = sort classes of some $ of like operators Def 'm ( X ) of C(X7 modules Ele > ~ via - - theory Kasparov Def 'm KK by nd of . More definitions geometric : - Douglas KBL Baum ( theory x ) - ' definition Stolz quotient of spine - via a . bordismn by suitable bovdis of burden classes - ( theory for KO for ) out carried only so - ,

( Baum If Douglas theory ) . . k*BD( X ) Set f) of classes ( equivalence = M E , , ' M uefd closed spin • a - f M X : → continuous . map E bundle vector M over . The by relation grnrated equivalence is ' bondisun spin - - of direct bundles vector Sum - : ( ) f ( ) fltf M En Ez MIM E to Ez I ~ , , , , , modification Spherical - ( takes ) wheel periodicity Bott can of

Spheriae reodificahoni vbe Given f ) spine 2k CM M W W di E = → - - , , , , ST Su the Define F where dual of = is : , W bundle reduced vertical spiuor IT 5 ( r ) M Wto over → . T*E ) fot ( f ( two Then C Weil M E S 1 ~ , , , , - Douglas ) Thun ( Baum Baum Higson Schick - - , - 1982 2007 KB*DC× ) ) K ex ± *

Using the Atiyeh orientation ' Aspin K - * : K T obtains one canonical a map szsrfitcx K C * ) K* x ) ) ( a → * Spine R * ) ( Hopkins Thu Hovey : - 1992 hououorpleisn ( k ) The isomorphism is an .

We triangle have thus of isomorphism a ±→ R5*Pih{ xlxosqgs K*BD(X7 . K* ±\ 1± k*C× ) Note horizontal The by induced : arrow is : tixsk Mn Sk ) , prz*E , fopr , ) ( ( f) ( 1- E • → , Thus this it cheat somewhat is surprising wrap is surjeehve .

§ 2 Twisted analogues Assume M spice not is : , K* for have Poincare deeaeitg - no have R Dirac operator canonical - no on . twisting General ~S out way : s Recall spins definition structure of i - Ual Spins so → → - " gti ↳ B spine B B Bleep 13441 So . > - → → - # " r Kc 743 ) . , n

t Put B BBUU ) let B X be = → a eeeap . . Def c A twisted structure for oriented Spin ae . to rnaucfoed write X represented by map is a stable oh isomorphism vector bundles an 6 S prig 8 v → → t 1 1 pr , B Bso X So → M - x K the isouoopheisns such Two provide same f it they twisted differ by structure - a ( ( ' isomorphism ) omit the details spin we - .

I B ) ( twisted Def for spine X bordiseee - , c Askin ( ) X ; twisted spine Gordis T = - - ) ( , f classes :n→X M of pairs Str twisted spine with - a . Remark introduce to There other also i ways are twisted ' the bordism and nomore spin of - twisted spin structure a . Them different to also ways several are two K ( X ; T ) K theory introduce shed groups - * for X B with reference space → a a map -

there Tu twisted spin ' Aliyah is any case ; ) ( lronuomorptris by index given an - RsP*in{ K ( ) ) × X - ;t ;e * factors through hoeeorplee of and a r*4i" ( X ) K ( it Xitl - * 9 1 Oxrsynikx a rsrfi ' ( . .tl x

Def ( ) D homology broke - . Wang 2013 Baum Carey - - , DB k ( ) Set ( of Xit ,E ,f ] classes M = equivalence * manifold closed M with ° a map f X or → : twisted , f ) ( spine sbr for M . a - , bundle vector E M a over . The by equivalence relation generated is twisted spine bordisu . bundles of direct vector seem • spherical modifications • .

- Carey Remain Baum that Wang conjectured : - K¥3 ( X. K E ( e) ) × ;t * ( Thin ) Baum Khorauei ] Schick - - - . . triangle These of isomorphism is a ;K* ⇐ RsIinE× KIBCX ) e) ;t gay - ; he I ± K * ( X ) E ; Note would expect priori again : a one to horizontal the be Lonoreovplclse iujedive but be to not suvfeehwe .

Remark proof on : In theorem the three • one compares generalized theories defined homology which are , the fixed category of or spaces over a 13 BBUCE ) namely = space . e the theorem techniques we using ° prove , from topology In algebraic particular . this to the statements allows prove localized the individual primes at to stalemate the owhegrae in get order

§ Aspects 3 the proof the of twisted of of version theorem tlophius the Hovey the at 2 Prime - to of dive need into deep topology → algebraic sea Homology theories fixed for • over spaces a base through be B represented can space B B Eu : of → Sequence spaces a over •• theories transformations Natural of seed be by represented of Can maps sequences for have B Eu ' MEN Eu over - ; i e - / b B

to for In obtain representing order spaces x the Rs←Pin< ( X ; K*C Xi e) transformation e) natural → BBUCI for ) it useful to model concrete is use a U Wyse .µµ Def Put B BPUCH ) PUCH BPU ) = = = , . for Hills +1 separable t a space - . Rs←M5( The for I K En and the * C ) and spaces ; i , HQHUI ,±og,S transformation by natural roughly are given Xu " ( @ ten ) ( H PU → EPU ' * pufoedcecu , EPU xpu 11 11 KB e Mspinps n . ,n

M spin B. C The < Mspvnps spectrum Spaces yield a u the KB while KB spectrum spaces yield a , , ( Hebestseit Thun . ) , 7 K there Is eoeaeey : - a 2013 hornobopy B at 2 equivalence of spectra over Mspiutg ⇒ " " TO k 2- , ] Partition where KB KB the connected of So-called 0 cover • is - " I formal denotes • suspension ( ) Petersen Brown parameterized Anderson Splitting - - -

Remark the from One parameterized : see can that Anderson splitting Brown Peterson of Copy a - ( ) theory splits but Connected K off parameterized - , this additive result yields igush an . Hopkins the and Hovey Used paraemebnzd we Brown the Anderson Peterson of splitting version - - to Mspief construct resolution at sort of of 2 a - theory K to Hopkins there the Hovey prone - the is Setting Their parameterized Un argument . and be carried of algebraic nature is over can .

§4 Sketch he twisted of of the of proof version - Hovey Hopkins theorems from 2 away we buried : on f*EP4 ( Kloraaei ) Thin let P= BPU E :X → : . , 2011 Then there isomorphism is a a => P ) k K ( Q K* ( ) Xit * * K*( PU ) PU ) K*( P ) K* the where multi ( acts by on - P the P action by PU × Plo induced → canon , lw oinorpceisn the K*( Ka PUT and aebs una - on Bna fix Kx PU ) K*K K ( K ( ) ) → → = * *

- Hovey the theorem Using Hopkins untwisted get we Kate CI szsxpiicp ) : K * ( e) @ × ; srsrfiicpu ) ] ^ 112 ( ns :⇒p¥e±k*h9÷emod¥ rs¥ ' 112 k ( P ) K Q * * K CPU ) *

* f the let P X be Observation IT EPU = → : ; X then PU projection the 5 of lendle over - factorisation have we a : rstriicp ) K* K+C×;t a ) → rs*Pid( ) Pu ^ t ) ± Ossme i. r*smt( P ) k Eog @ ; X ] rs*Piu' Pu ) ( I + rsrxiicxit K* ) *

Crucial for proof from the lemma 2 away [ I ] the let 2 then below A composition = . split " is surjechwe . szsrfitcp ) rs*Pit( I ) P 01L tot a @ , eotloxryanqpu * ) thin 's . e) surjeelioin → I × a a , Remark the howomorphe.su At 2 : RMF ' rtfiicx II. ) ( P .t - ; , , not in general a is . * ) from The Bspin B BUCI ) 2 e So × argument away uses :

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