Abstract

Abstract. The irreducible representations of the complex symplectic group of genus g are indexed by nonincreasing sequences of integers λ = (λ1 ≥ λ2 ≥ · · ·) with λk = 0 for k> g. A recent result of N.V. Ivanov implies that for a given partition λ, the cohomology group of a given degree of the mapping class group of genus g with values in the representation associated to λ is independent of g if g is sufficiently large. We prove that this stable cohomology is the tensor product of the stable cohomology of the mapping class group and a finitely generated graded module over Q[c1,..., c |λ|], where deg(ci) = 2i and |λ | = ∑ i λi. We describe this module explicitly. In the same sense we determine the stable rational cohomology of the moduli space of compact Riemann surfaces with s given ordered distinct (resp. not necessarily distinct) points as well as the stable cohomology of the universal Abel–Jacobi map. These results take into account mixed Hodge structures. 1.

Citations

96	The second homology of the mapping class group of an orientable surface, Inventiones 72 - Harer - 1983
86	Representatin Theory”, Graduated text - Fulton, Harris
65	Théorème de Lefschetz et critères de dégénérescence de suites spectrales - Deligne - 1968
45	Symmetric products of an algebraic curve, Topology 1 - Macdonald - 1962
26	Complexes of curves and the Teichmüller modular group, Uspekhi Mat. Nauk 42 - Ivanov - 1987
26	On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients, Mapping class groups and moduli spaces of Riemann surfaces - Ivanov - 1991
8	Improved stability for the homology of the mapping class groups of surfaces, Duke University preprint - Harer - 1993