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

Tara Brendle, Nathan Broaddus, Andrew Putman

Published 2020-03-24Version 1

Generalizing work of Fullarton-Putman in the closed case, we give two proofs that appropriate finite-index subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. 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. As an application, we prove upper and lower bounds on the coherent cohomological dimension of the moduli space of curves with marked points, and in particular compute this coherent cohomological dimension for $g \leq 5$ and any number of marked points.