The stable mapping class group and Q(CP_+~∞)

摘要

In [T2] it was shown that the classifying space of the stable mapping class groups after plus construction Z * BΓ_∞~+ has an infinite loop space structure. This result and the tools developed in [BM] to analyse transfer maps, are used here to show the following splitting theorem. Let Σ~∞(CP_+~∞)_P~＾ approx= E_0 V…VE_(p-2) be the "Adams-splitting" of the p-completed suspension spectrum of CP_+~∞. Then for some infinite loop space W_p, (Z * BΓ_∞~+)_p~＾ approx = Ω~∞(E_0) * … * Ω~∞(E_(p-3)) * W_p where Ω~∞E_i denotes the infinite loop space associated to the spectrum E_i. The homology of Ω~∞E_i is known, and as a corollary one obtains large families of torsion classes in the homology of the stable mapping class group. This splitting also detects all the Miller-Morita-Mumford classes. Our results suggest a homotopy theoretic refinement of the Mumford conjecture. The above p-adic splitting uses a certain infinite loop map α_∞: Z * BΓ_∞~+ → Ω~∞CP_(-1)~∞ that induces an isomorphims in rational cohomology precisely if the Mumford conjecture is true. We suggest that α_∞ might be a homotopy equivalence.