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 × [special characters omitted] and Y × [special characters omitted] 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 × [special characters omitted] and N × [special characters omitted] 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 × [special characters omitted] and N × [special characters omitted] 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 × [special characters omitted] and N × [special characters omitted] 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.