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Calculating cohomology groups of moduli spaces of curves via algebraic geometry

Summary: May 1998
Calculating cohomology groups of moduli spaces of curves
via algebraic geometry
Enrico Arbarello and Maurizio Cornalba 1
In this paper we compute the first, second, third, and fifth rational cohomology
groups of M g;n , the moduli space of stable n­pointed genus g curves. It turns out that
H 1 (M g;n ; Q ), H 3 (M g;n ; Q ), and H 5 (M g;n ; Q) are zero for all values of g and n, while
H 2 (M g;n ; Q) is generated by tautological classes, modulo relations that can be written
down explicitly; the precise statements are given by Theorems (2.1) and (2.2). We are
convinced that the computation of the fourth cohomology of all moduli spaces M g;n should
also be accessible to our methods.
It must be observed that some of these results are not new. In fact, it is known that
M g;n is simply connected (cf. [2], for instance), while Harer has determined H 2 (M g;n ; Q)
[8]; once this is known, it is not hard to compute the corresponding group for M g;n . Harer
[10] has also shown that H 3 (M g;n ; Q) vanishes, at least for large enough genus. What
is really new here is the method of proof, which is based on standard algebro­geometric
techniques, rather than geometric topology. Especially for odd cohomology, this provides
proofs that are quite short and, we hope, rather transparent. It should also be noticed
that the odd cohomology of M g;n , at least in the range we can deal with, seems to be
somewhat better behaved than the one of M g;n , for it is certainly not the case that the


Source: Arbarello, Enrico - Dipartimento di Matematica, Universitą di Roma "La Sapienza"


Collections: Mathematics