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 \S 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 Mumford-Morita-Miller classes. The details of the results sketched in this note will appear elsewhere.

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Published 1 January 1996