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.