The sphericity of the complex of non-degenerate subspaces

TL;DR

This paper proves that the complex of non-degenerate subspaces in a finite-dimensional vector space with a sesquilinear form is homotopy equivalent to a wedge of spheres, with implications for arithmetic groups and finite groups of Lie type.

Contribution

It establishes the sphericity of complexes related to non-degenerate subspaces and generalized Phan geometries, extending understanding of their topological and algebraic properties.

Findings

Complexes are homotopy equivalent to wedges of spheres

Sphericity implies finiteness properties of arithmetic groups

Supports group-theoretical recognition of finite Lie type groups

Abstract

We prove that the complex of proper non-trivial non-degenerate subspaces of a finite-dimensional vector space endowed with a non-degenerate sesquilinear form is homotopy equivalent to a wedge of spheres. Additionally, we show that the same is true for a slight generalization, the so-called generalized Phan geometries of type A_n. These generalized Phan geometries occur as relative links of certain filtrations. Their sphericity implies finiteness properties of suitable arithmetic groups and allows for a revision of Phan's group-theoretical local recognition of suitable finite groups of Lie type with simply laced diagram.