An analogue of Rognes' connectivity conjecture for free groups

TL;DR

This paper proves that the common basis complex of a free group has the homotopy type of a wedge of spheres, extending Rognes' connectivity conjecture from linear groups to free groups.

Contribution

It establishes an analogue of Rognes' connectivity conjecture for free groups and provides multiple homotopy-equivalent models of the common basis complex.

Findings

Common basis complex has homotopy type of a wedge of spheres of dimension 2n-3

Provides models in terms of free factors and sphere systems

Extends connectivity conjecture to free groups

Abstract

We show that the common basis complex of a free group of rank $n$ has the homotopy type of a wedge of spheres of dimension $2n-3$. This establishes an $\mathrm{Aut}(F_n)$-analogue of the connectivity conjecture that Rognes originally stated for $\mathrm{GL}_n(R)$. To prove this, we provide several homotopy-equivalent models of the common basis complex, both in terms of free factors in free groups and in terms of sphere systems in 3-manifolds.