Real К-theory j87

This IS equivalent (mod/J to (i ag){l a А)^,{р''a1)ocj We saw in the proof of Lemma 2 that v annihilates /^ Thus '^

( lAg ) ( i ; ^a ) = (lAg)(lA^)i;X(/Al)a,

J

If we let a = ^ (p'' л 1) aj, we find that there is a ^8 such that J

( lAf ) ß = v4-(lAÄ)V(x'

Write

ß^J ] { / Al ) ßj { modIJ J

Again , we are working mod 2, so

( 1 Af)ivß) = Y,{p'Al){l Af)pßj = {lAA)il A/)iß(m0d /J

Thus (1 Af){vß)-{1 A Л){1 Af)ß is i;-torsion Hence we get that

v^a - { lAA ) { v^ ( x' + {iAf))ß

is y-torsion, completing the proof П

Note that if the Anderson-Brown-Peterson sphttmg could be made into a fco-module spectrum splitting (which it can not be [KS]), Theorem 3 would be obvious since then we would have t;"^MSpmc^ \/ КО It can be shown

\ФJ using [St] that the homotopy groups of t;~^MSpin are what they should be for such an equivalence to hold

Corollary 1 For any ring spectrum R,

( MSpin A R)^ (X) (8)(MSp,n л R).(KO A R)^

= (Li MSpin A R)^ [X) (X)(^^ Mspm . Rb(KO A R)^

Also

MSpm ; {X) ®м8рш.^K^ = Li MSpin; (X) ®ь, MSp.„j K^ Proof Since p is a unit in (КО a R)^ ,

( Li MSpm A R)^ (X) ®^^^ Mspm л яь(КО a R)^

= {v-'{L, MSpm aR)),(X)(x)(, .(^^^м8ршлК))ДКО aR)^

Similarly ,

( MSpin A R)^ (X) (8)(MSp,„ . r),(KO a R),

= (t;-4MSpm A R))^(X)®,, чм8ршлК))ЛК0 a R)^