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

Zhong, Xiyan

doi:10.2140/gt.2026.30.155

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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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

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Volume 30, issue 1 (2026)