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

By Benjamin Brück, Radhika Gupta. Proceedings of the London Mathematical Society, 2018

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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.

Subfields: Geometry and Topology, Mathematical Physics

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