Separable Homology of Graphs and the Whitehead Complex

We introduce the "Whitehead complex", a one-complex associated to a finite regular cover of the rose and show that it is connected if and only if the fundamental group of the associated cover is generated by its intersection with the set of elements in proper free factors of the fundamental group of the rose. The Whitehead complex admits an action of the outer automorphism group of the fundamental group of the rose by isometries if the associated cover corresponds to a characteristic subgroup. We prove that the Whitehead complex of the rose has infinite diameter and is nonhyperbolic, implying it is not quasi-isometric to the free splitting complex or the free factor complex. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]