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

In [T2] it was shown that the classifying space of the stable mapping class groups after plus construction Z x B Gamma (+)(infinity) 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 Sigma (infinity)(CP+infinity)(p)(Lambda) similar or equal to E(0)nu (...)nuE(p-2) be the "Adams-splitting" of the p-completed suspension spectrum of CP+infinity. Then for some infinite loop space W-p, (Z x B Gamma (+)(infinity))(p)(Lambda)similar or equal to Omega (infinity)(E-0) x(...)x Omega (infinity)(Ep-3) x W-p where Omega E-infinity(i) denotes the infinite loop space associated to the spectrum Ei. The homology of Omega (infinity) 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 a(infinity) : Z x B Gamma (+)(infinity) --> Omega (infinity) CP-1infinity that induces an isomorphims in rational cohomology precisely if the Mumford conjecture is true. We suggest that a,,, might be a homotopy equivalence.