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

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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 ≤ 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 ≥ 1 marked points, these bounds turn out to be independent of n, and depend only on the genus.

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Acknowledgments

We thank Joan Birman for her contributions to an earlier collaboration, funded in part by the Simons Foundation, which helped to lay the groundwork for this paper. We additionally thank MSRI for its support for the first and second authors in this early phase. We would also like to thank Benson Farb and the University of Chicago for support and funding; much of the work on this paper was completed while the first author was a visitor there. Finally, we would like to thank Eric Riedl and Chris Schommer-Pries for helpful conversations.

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Authors and Affiliations

  1. School of Mathematics & Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ, UK

    Tara Brendle

  2. Department of Mathematics, Ohio State University, 231 W. 18th Ave., Columbus, OH, 43210, USA

    Nathan Broaddus

  3. Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN, 46556, USA

    Andrew Putman

Authors
  1. Tara Brendle
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  2. Nathan Broaddus
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  3. Andrew Putman
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Correspondence to Nathan Broaddus.

Additional information

Supported in part by NSF grant DMS-1811210.

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Brendle, T., Broaddus, N. & Putman, A. The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2566-9

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  • DOI: https://doi.org/10.1007/s11856-023-2566-9

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