The stable mapping class group and Q(ℂP∞+)

Abstract.

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 ≃E ₀∨...∨E _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 ≃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.