International Congress of Mathematicians

454 U. Tillmann7. Splittings and (co) homological resultsThe main result of [8] is a partial splitting of the compositionUJ o a : Z x FF+ —• Q(P°?) ~ Q(S°)x Q(P°°).This is achieved by constructing a map ß from P + to Z x FF+ and then extendingit to the free infinite loop space Q(W°^) utilizing the infinite loop space structureon Z x FF+. In order to construct ß, approximate P°° ~ BS 1 by the classifyingspaces of cyclic groups Cp» for n —¥ oo, one prime p at a time, as the cyclic groupscan be mapped into suitable mapping class groups. However, this means that wehave to work with p-completions.Let Yp A denote the p-completion of Y and g £ Z p be a topological generatorof the p-adic units (g = 3 if p = 2). Denote by fi k : P°° —¥ (V°°) p the map thatrepresents k times the first Chern class in H 2 (W°°,Z P ).Theorem 7.1. [8]. There exists a map ß : (Q(S°) x Q(P°°))£ -• (Z x FF+ )£such thatÜ O Ö °^("o 2 l-gi/j*The map 1 — gfi 9 induces multiplication by 1 — g n+1 on H 2n (^°°;Z^) whichis a p-adic unit precisely if n ^ —1( mod p — 1). The following applications ofTheorem 7.1 are also found in [8]. There is a splitting Q(W°^) p ~ F 0 x • • • x F p _ 2corresponding to the idempotent decomposition of Z[Z/p x ] c Z P [Z*].Corollary 7.2. For some W p , there is a splittingof infinite loop spaces(Z x FF+ )£ ~ F 0 x • • • x F p _ 3 x W p .The Z/p-homology of Q(P+) is well-understood in terms of Dyer-Lashof operation.These are homology operations for infinite loop spaces that are formallysimilarto the Steenrod operations. For each generator a, £ H 2 i^°° = Z there isan infinite family of Z/p-homology classes freely generated by the Dyer-Lashof operations.The product F 0 x • • • x F p _ 3 contains precisely those families for whichi ^ ^l(mod p — 1), giving a huge collection of new p-torsion in H^BY^.For odd primes p, Madsen and Schlichtkrull [MS] found split surjective mapslo and l_i of infinite loop spaces such that the following diagram is commutativeQ°°Th(^F)£ — ^Q(P^)£19V , in, ,, nrnA(Z x BU)$ — ^ (Z x BU)'