Classifications of 2-complexes whose finite fundamental group is that of a 3-manifold

Abstract

We consider spines of spherical space forms; i.e., spines of closed oriented 3-manifolds whose universal cover is the 3-sphere. We give sufficient conditions for such spines to be homotopy or simple homotopy equivalent to 2-complexes with the same fundamental groupGand minimal Euler characteristic 1. If the group ring ℤGsatisfies stably-free cancellation, then any such 2-complex is homotopy equivalent to a spine of a 3-manifold. IfK1(ℤG) is represented by units andKis homotopy equivalent to a spineX, thenKandXare simple homotopy equivalent. We exhibit several infinite families of non-abelian groupsGfor which these conditions apply.