The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary

Abstract

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.Comment: 29 pages, 3 figures; some small correction