Prym representations and twisted cohomology of the mapping class group with level structures

Zhong, Xiyan

doi:10.2140/gt.2026.30.155

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Prym representations and twisted cohomology of the mapping class group with level structures

Xiyan Zhong

Geometry & Topology 30 (2026) 155–202

DOI: 10.2140/gt.2026.30.155

Abstract

We compute the twisted cohomology of the mapping class group with level structures, with coefficients in the $r$ -tensor powers of the Prym representations for any positive integer $r$ . When $r \geq 2$ , we show that the cohomology exhibits instability for large genus, whereas it remains stable for $r = 0$ or $r = 1$ . As a corollary, we prove that the symplectic Prym representation associated with any finite abelian regular cover of a nonclosed finite-type surface is infinitesimally rigid.

Keywords

mapping class group, level structures, Prym representation

Mathematical Subject Classification

Primary: 20F05, 20J06, 57K20, 57S05

References

Publication

Received: 3 April 2024

Revised: 17 April 2025

Accepted: 28 May 2025

Published: 19 January 2026

Proposed: Dan Abramovich

Seconded: Benson Farb, Nathalie Wahl

Authors

Xiyan Zhong
Department of Mathematics University of Notre Dame South Bend, IN United States

This article is currently available only to readers at paying institutions. If enough institutions subscribe to this Subscribe to Open journal for 2026, the article will become Open Access in early 2026. Otherwise, this article (and all 2026 articles) will be available only to paid subscribers.

Volume 30, issue 1 (2026)