# Tangential stabilization of spherical spaceforms

This dissertation is devoted to the study of the question of stable equivalence. That is, given two nonhomeomorphic topological spaces X and Y, does there exist some integer k > 0 such that the products X x Rk and Y x Rk are homeomorphic. In the case of closed manifolds M and N it is a known result that there exists a k ≥ 0 such that M x Rk and N x Rk are homeomorphic if and only if M and N are tangentially homotopy equivalent (i.e. there is a homotopy equivalence f : M → N such that the pullback of the stable tangent bundle of N is the stable tangent bundle of M). Therefore, given two tangentially homotopy equivalent manifolds M and N, we ask: what is the least value of k ≥ 0 such that M x Rk and N x Rk are homeomorphic? Qualitatively, we describe results to such an optimal value question in terms a concept called tangential thickness, loosely defined to be the least k ≥ 0 such that M x Rk and N x Rk are homeomorphic. In our analysis, we will consider the tangential thickness of spherical spaceforms; manifolds of the form S n/G for G a finite group acting freely on Sn. If the group action is linear, we call the resulting manifold a linear spherical spaceform. If the group action is nonlinear, we call the resulting manifold a fake spherical spaceform. Specifically, we will consider the case of quaternionic spaceforms in which the group G is the generalized quaternion group First, we shall show that the tangential thickness of linear quaternionic spaceforms is 3. In the case of fake quaternionic spaceforms, one can have varying thicknesses. Thus, we shall classify those fake quaterionic spaceforms with tangential thickness 1, 2 or 3. We shall also prove the existence of fake quaternionic spaceforms with tangential thickness ≥ 4