The primary approximation to the cohomology of the moduli space of curves and cocycles for the stable characteristic classes

The purpose of the present note is to announce our recent results on the cohomology of the moduli space of curves or equivalently ( over the rationals) the cohomology of the mapping class group of orientable surfaces. Our main results are twofold. First we construct explicit group cocycles for any of the known stable characteristic classes ( the Mumford-Morita-Miller classes) of the moduli spaces. Secondly, by combining our result with that of Hain in [H2), we show that the "continuous part" of the cohomology of the moduli space (see §5 for the definition) is exactly equal to the subalgebra generated by the above stable classes. This second result may be considered as a supporting evidence for the conjecture that the stable cohomology of the moduli spaces would be equal to the polynomial algebra generated by the Mumford­Morita-Miller classes. The details of the results sketched in this note will appear elsewhere

Riemann-Hurwitz formula for Morita-Mumford classes and surface symmetries

Let a finite group G act on a compact Riemann surface C in a faithful and orientation preserving way. Then we describe the Morita-Mumford classes en (Ca) E H2n (G;Z) of the homotopy quotient (or the Borel construction) Ca of the action in terms of fixed-point data. This fixed-point formula is deduced from a higher analogue of the classical Riemann-Hurwitz formula based on computations of Miller [Mi] and Morita [Mo]

On the stable cohomology algebra of extended mapping class groups for surfaces

Let :E0,1 be an oriented compact surface of genus g with 1 boundary component, and r g,1 the mapping·class group of :Eg,1 · We determine the stable cohomology group of r 0,1 with coefficients in H1 (:Eg,1; Z)®n, n l, explicitly modulo the stable cohomology group with trivial coefficients. As a corollary the rational stable cohomology algebra of the semi-direct product r g,1 1>< H1 (:E0,1; Z) (which we call the extended mapping class group) is proved to be freely generated by the generalized Morita-Mumford classes in;J 's (i 0, j 1, i + j 2) [Ka) over the rational stable cohomology algebra of the group r g,1

The meromorphic solutions of the Bruschi-Calogero equation

We give all the meromorphic functions defined near the origin 0 E C satisfying a functional equation investigated by Bruschi and Calogero [1], [2]

The mapping class group and the Meyer function for plane curves

For each d>=2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d=4, using our Meyer function, we can define the local signature for 4-dimensional fiber spaces whose general fibers are non-hyperelliptic compact Riemann surfaces of genus 3. Some computations of our local signature will be given.Comment: 24 pages, typo adde

On the stable cohomology algebra of extended mapping class groups for surfaces

Let :E0,1 be an oriented compact surface of genus g with 1 boundary component, and r g,1 the mapping·class group of :Eg,1 · We determine the stable cohomology group of r 0,1 with coefficients in H1 (:Eg,1; Z)®n, n l, explicitly modulo the stable cohomology group with trivial coefficients. As a corollary the rational stable cohomology algebra of the semi-direct product r g,1 1>< H1 (:E0,1; Z) (which we call the extended mapping class group) is proved to be freely generated by the generalized Morita-Mumford classes in;J 's (i 0, j 1, i + j 2) [Ka) over the rational stable cohomology algebra of the group r g,1