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

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 $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$.