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 |
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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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