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

Abstract

We show that the complex of free factors of a free group of rank n >= 2 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n>=2, 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 spitting complex is contractible.