Spaces of Spheres and the Torelli Subgroup of Out(F_n)

Using ideas from 3-manifolds, Hatcher--Wahl defined a notion of automorphism groups of free groups with boundary. We study their Torelli subgroups, adapting ideas introduced by Putman for surface mapping class groups. We show that these groups are finitely generated, and also that they satisfy an appropriate version of the Birman exact sequence. We also define a version of the cycle complex, first introduced by Bestvina, Bux, and Margalit, on which the Torelli subgroup of the outer automorphism group of a free group acts. Our final main result is that this complex is contractible.