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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 splitting complex is contractible.

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