On the rational cohomology of spin hyperelliptic mapping class groups

2024/01/06 by Gefei Wang, Wang, Gefei
Mathematics · #14H30 (Secondary) #20F36 (Primary) #20J06 #57K20 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2401.03110

openalex publication_date 2024/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathfrakG be the subgroup \mathfrakS_n-q × \mathfrakS_q of the n-th symmetric group \mathfrakS_n for n-q ≥ q. In this paper, we study the \mathfrakG-invariant part of the rational cohomology group of the pure braid group P_n. The invariant part includes the rational cohomology of a spin hyperelliptic mapping class group of genus g as a subalgebra when n=2g+2, denoted by H^*(P_n)^\mathfrakG. Based on the study of Lehrer-Solomon, we prove that they are independent of n and q in degree *≤ q-1. We also give a formula to calculate the dimension of H^*(P_n)^\mathfrakG and calculate it in all degree for q≤ 3.

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