Homology of the mapping class group for surfaces of genus 2 with boundary

TL;DR

This paper computes the integral homology of the mapping class group for genus 2 surfaces with boundary and punctures, focusing on cases where 2g + m < 6, including genus 2 with no or one puncture.

Contribution

It provides the first explicit homology computations for genus 2 surfaces with boundary and punctures in the specified range, filling a gap in the understanding of these groups.

Findings

Homology groups computed for genus 2 surfaces with boundary and punctures.

Results include cases with no punctures and one puncture.

Provides foundational data for further topological and geometric studies.

Abstract

We report on the computation of the integral homology of the mapping class group of genus g surfaces with one boundary curve and m punctures, when 2g + m is smaller than 6. In particular, it includes the genus 2 case with no or one puncture.