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 > 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.

29 pages, 7 figures; v2: replaced a reference which allowed to shorten some proofs; v3: edited according to referee's suggestions, in particular significantly shortened the proof of Theorem C (contractibility of relative free splitting complexes) using folding paths; to appear in Proceedings of the London Mathematical Society