Homological Stability of automorphism groups of quadratic modules and manifolds

TL;DR

This paper establishes homological stability for automorphism groups of quadratic modules and manifolds, extending previous results to more general settings including singular forms and virtually polycyclic fundamental groups.

Contribution

It proves homological stability for general linear and unitary groups over rings with finite stable ranks, without assuming well-behaved modules or quadratic forms, and extends stability results to certain manifold moduli spaces.

Findings

Homological stability for linear and unitary groups over rings with finite stable rank.

Extension of stability results to manifolds with virtually polycyclic fundamental groups.

Applicability to quadratic modules with singular forms.

Abstract

We prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules and quadratic modules to be well-behaved in any sense: for example, the quadratic form may be singular. This extends results by van der Kallen and Mirzaii--van der Kallen respectively. Combining these results with the machinery introduced by Galatius--Randal-Williams to prove homological stability for moduli spaces of simply-connected manifolds of dimension 2n > 4, we get an extension of their result to the case of virtually polycyclic fundamental groups.