Homology stability for symplectic groups

TL;DR

This paper proves homology stability for symplectic groups over rings with finite stable rank, using a nerve theorem to analyze the connectivity of certain posets related to isotropic vectors.

Contribution

It introduces a nerve theorem for posets and applies it to establish homology stability for symplectic groups, confirming a conjecture by Charney.

Findings

Poset of isotropic vectors is highly connected

Homology stability for symplectic groups over rings with finite stable rank

Validated Charney's conjecture from the 1980s

Abstract

In this paper the homology stability for symplectic groups over a ring with finite stable rank is established. First we develop a `nerve theorem' on the homotopy type of a poset in terms of a cover by subposets, where the cover is itself indexed by a poset. We use the nerve theorem to show that a poset of sequences of isotropic vectors is highly connected, as conjectured by Charney in the eighties.