Homotopy type of the complex of free factors of a free group

TL;DR

This paper establishes the homotopy equivalence of the complex of free factors of a free group to a wedge of spheres, and explores related complexes' topological properties, advancing understanding of free group automorphisms.

Contribution

It proves the homotopy types of several complexes associated with free groups, including the free factor complex and free splitting complex, revealing their topological structures.

Findings

The complex of free factors is homotopy equivalent to a wedge of spheres.

The complement of Outer space in the free splitting complex is (n-2)-connected.

Relative free splitting complexes are contractible.

Abstract

We show that the complex of free factors of a free group of rank n > 1 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n > 1, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent to the complex of free factor systems and moreover is (n-2)-connected. In addition, we show that for every non-trivial free factor system of a free group, the corresponding relative free splitting complex is contractible.