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UBC Theses and Dissertations

The spherical space form problem Murray, David William


This thesis is intended to be a fairly complete account of the spherical space form problem - both in its classical sense and its more general modern formulation. In 1891, Killing posed the Clifford-Klein spherical space form problem, essentially as follows: Find and classify up to isometry all complete, connected, Riemannian manifolds of constant positive curvature. In 1925, Hopf showed this problem is equivalent to finding all finite groups G which can act fixed point freely on spheres by isometries. A natural generalization of this problem is obtained by relaxing the condition that G act by isometries; for example, we may seek free topological actions of finite groups on spheres, the so-called "topological spherical space form problem". An explicit solution of the Clifford-Klein spherical space form problem was first obtained by Wolf in his book "Spaces of Constant Curvature". After a preliminary .chapter on the relationship between group cohomology and free actions of groups on spheres, we discuss Wolf's work in chapter 2. We then turn to the topological spherical space form problem, and present a "homotopy version" due to Swan in chapter 3. In chapter 4, we give a necessary condition for free topological actions due to Milnor, and briefly discuss some pertinent work of Petrie and of Lee. In the fifth and final chapter, we discuss work of Madsen, Thomas, and Wall which determines sufficient conditions for free actions of finite groups on spheres. These conditions are of a type frequently encountered in classical group theory; their effect is to limit the types of subgroups which G may possess.

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