A finite presentation for the automorphism group of the first homology of a non-orientable surface over $\mathbb Z_2$ preserving the mod $2$ intersection form

TL;DR

This paper provides a finite presentation for the automorphism group of the first homology of a non-orientable surface over Z2 that preserves the mod 2 intersection form, and computes its homology groups.

Contribution

It introduces a finite presentation for the automorphism group and calculates its first and second homology groups, advancing understanding of its algebraic structure.

Findings

Finite presentation of the automorphism group obtained.

First and second homology groups computed.

Enhanced understanding of the algebraic structure of the automorphism group.

Abstract

Let $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$ be the group of automorphisms on the first homology group with $\mathbb Z_2$ coefficient of a closed non-orientable surface $N_g$ preserving the mod $2$ intersection form. In this paper, we obtain a finite presentation for $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$. As applications we calculate the first homology group and the second homology group of $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$.