International Congress of Mathematicians

Strings and the Stable Cohomology of Mapping Class Groups 455Corollary 7.3. For odd primes p and some space V p , there is a splitting of spaces(FF+ )£ ~ BU£ x V p .This gives a Z p -integral version of Miller and Morita's theorem: the polynomialalgebra Z p [ci,c 2 ,...] is a split summand of H*(BY 00 ; Z p ). The divisibility of thetautological classes K, at odd primes p can also be deduced from the above diagram.Corollary 7.4. If i = —l(modp— 1), then K, is divisible by p 1 +"p(*+ 1 ) where v p isthe p-adic valuation. Otherwise, p does not divide K,.In the light of [9] this result is sharp.8. Geometric interpretationa : FF+ —t Qg°Th(^F) is a homotopy equivalence if and only if it inducesan isomorphism in oriented cobordism theory Qf°. An element in Q|°(FF+) =0|°(FF 0O ) is a cobordism class of oriented surface bundles F —t E n+2 ^y M n . Anelement in Q|°(Qg°Th(^F)) is a cobordism class of pairs [n : E n+2 —t M n ,n] ofsmooth maps n and stable bundle surjections from FF to TT*TM. (Upto cobordismone can assume that n is a vector bundle surjection.) a maps a bundle [F —tE ^y M] to the pair [n : E —t M, Dn] where Dn denotes the differential of n.Hence, a is a homotopy equivalence if and only if each cobordism class of pairs[n : E n+2 —t M n ,n] contains a "unique" representative with n a submersion.It is this geometric formulation that underpins the solution to the Mumfordconjecture by Madsen and Weiss. A key ingredient of the proof is the Phillips-Gromov ft-principle of submersion theory: A pair (g : X —t M, g : TX —t g*TM)can be deformed to a submersion - provided X is open. F above, however, is closed.The approach taken in [9] is to replace n : E —t M by g = nopn : X = E x R —t M.Now the submersion ft-principle applies and g can be replaced by a submersion /.The proof then consists of a careful analysis of the singularities of the projectionpi'i : X —t R on the fibers of /. At a critical point it uses Harer's Stability Theorem2.1.is a homo­Madsen-Weiss Theorem 8.1. The map a : Z x FF+ -¥ ii°°Th(-L)topy equivalence.References[1] C.-F. Bödigheimer & U. Tillmann, Stripping and splitting decorated mappingclass groups, Birkhäuser, Progress in Math. 196 (2001), 47^ 57.[2] C.J. Earle & J. Eells, A fibre bundle description of Teichmüller theory, J. Diff.Geom. 3 (1969), 19-13.[3] S. Galatius, Homology of Q°°SCF~ and Q°°CF~ , preprint 2002.[4] J.L. Harer, Stability of the homology of the mapping class groups of orientablesurfaces, Annals Math. 121 (1985), 215^249.