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.

arXiv comment29 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

References in corpus (6)

• Geometry of the Complex of Curves I: Hyperbolicity
• Approximation of definable sets by compact families, and upper bounds on homotopy and homology
• Relative free splitting and free factor complexes I: Hyperbolicity
• Contractibility of deformation spaces of G-trees
• On the bordification of outer space
• Between buildings and free factor complexes: A Cohen-Macaulay complex for Out(RAAGs)

Cited by in corpus (4)

• Between buildings and free factor complexes: A Cohen-Macaulay complex for Out(RAAGs)
• The free factor complex and the dualizing module for the automorphism group of a free group
• Cohen--Macaulay Complexes, Duality Groups, and the dualizing module of $\mathrm{Out}(F_N)$
• Connectivity of partial basis complexes of freely decomposable groups