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Register a new userThe high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary
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We give two proofs that appropriately defined congruence subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. In particular we give bounds that are super-exponential in each of three variables: number of punctures, number of boundary components, and genus, generalizing work of Fullarton-Putman. Along the way, we give a simplified account of a theorem of Harer explaining how to relate the homotopy type of the curve complex of a multiply-punctured surface to the curve complex of a once-punctured surface through a process that can be viewed as an analogue of a Birman exact sequence for curve complexes. As an application, we prove upper and lower bounds on the coherent cohomological dimension of the moduli space of curves with marked points. For $g leq 5$, we compute this coherent cohomological dimension for any number of marked points. In contrast to our bounds on cohomology, when the surface has $n geq1$ marked points, these bounds turn out to be independent of $n$, and depend only on the genus.
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Let $Gamma$ be a finite-index subgroup of the mapping class group of a closed genus $g$ surface that contains the Torelli group. For instance, $Gamma$ can be the level $L$ subgroup or the spin mapping class group. We show that $H_2(Gamma;Q) cong Q$ for $g geq 5$. A corollary of this is that the rational Picard groups of the associated finite covers of the moduli space of curves are equal to $Q$. We also prove analogous results for surface with punctures and boundary components.
In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Wittens conjectures, can be described completely algebraically as the homology of a certain differential graded Lie algebra. This two-parameter family is constructed by using a Lie cobracket on the space of noncommutative 0-forms, a structure which corresponds to pinching simple closed curves on a Riemann surface, to deform the noncommutative symplectic geometry described by Kontsevich in his subsequent papers.
For $4 mid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level $L$ subgroup of the mapping class group (in a previous paper, the author determined this rationally). This entails calculating the abelianization of the level $L$ subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of the mod $L$ symplectic group with coefficients in the adjoint representation.
The moduli space of flat SU(2) connections on a punctured surface, having prescribed holonomy around the punctures, is a compact smooth manifold if the prescription is generic. This paper gives a direct, elementary proof that the trace of the holonomy around a certain loop determines a Bott-Morse function on the moduli space which is perfect, meaning that the Morse inequalities are equalities. This leads to an attractive recursion for the Betti numbers of the moduli space, which agrees with the Harder-Narasimhan formula in the case of one puncture with holonomy -1.
The purpose of this paper is to describe an analogue of a construction of Costello in the context of finite-dimensional differential graded Frobenius algebras which produces closed forms on the decorated moduli space of Riemann surfaces. We show that this construction extends to a certain natural compactification of the moduli space which is associated to the modular closure of the associative operad, due to the absence of ultra-violet divergences in the finite-dimensional case. We demonstrate that this construction is equivalent to the dual construction of Kontsevich.
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