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

Brendle, T. , Broaddus, N. and Putman, A. (2023) The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary. Israel Journal of Mathematics, (doi: 10.1007/s11856-023-2566-9) (Early Online Publication)

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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.

Item Type:Articles
Status:Early Online Publication
Refereed:Yes
Glasgow Author(s) Enlighten ID:Brendle, Professor Tara
Authors: Brendle, T., Broaddus, N., and Putman, A.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Israel Journal of Mathematics
Publisher:Hebrew University Magnes Press / Springer
ISSN:0021-2172
ISSN (Online):1565-8511
Published Online:13 November 2023
Copyright Holders:Copyright © 2023, The Hebrew University of Jerusalem
First Published:First published in Israel Journal of Mathematics 2023
Publisher Policy:Reproduced in accordance with the publisher copyright policy

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