Homological stability for automorphisms of symmetric bilinear forms

TL;DR

This paper proves homological stability for automorphisms of symmetric bilinear forms over various rings, advancing understanding of the stable cohomology of odd orthogonal groups in low degrees.

Contribution

It establishes homological stability for automorphisms of symmetric bilinear forms over principal ideal domains, including fields and integers, and connects this to stable cohomology calculations.

Findings

Homological stability holds for automorphisms over specified rings.

Determination of large parts of the stable cohomology of odd orthogonal groups.

Results apply to rings like integers, Gaussian integers, and Eisenstein integers.

Abstract

We establish homological stability for automorphisms of symmetric bilinear forms over a class of principal ideal domains that includes all fields, the integers, the Gaussian integers, and the Eisenstein integers. In conjunction with Grothendieck-Witt theoretic calculations, this determines a large part of the stable cohomology of the odd orthogonal groups $O_{\langle g,g \rangle}(\mathbb Z)$ in low degrees.