Finite Abelian Group Actions on Orientable Circle Bundles over Surfaces

A finite group G acts freely on an orientable manifold M if each element of G is a homeomorphism of M, without fixed points, and the multiplication in G is the composition of homeomorphisms. The map from M to M/G of M to the orbit space is a regular cover map. Algebraically, associated with the G-action is a surjective homomorphism from the fundamental group of M into G. Two G-actions are equivalent if there exists an orientation preserving homeomorphism on M, inducing the identity on G, that preserves the group action. This topological definition is translated to an algebraic definition as two G-actions are equivalent if and only if the associated surjections into G are equivalent via an automorphism of the fundamental group of M. For the manifolds M considered in this dissertation every automorphism of the fundamental group of M can be realized by a homeomorphism of M. Hence there is a one-to-one correspondence between the topological and algebraic equivalence. The problem of classifying fixed-point free finite abelian group actions on surfaces had been investigated by, among others, Nielsen, Smith and Zimmermann. Nielsen classifies cyclic actions on surfaces. He gives a list of automorphisms which he uses in his classification. Smith does the classification for special abelian groups. His approach is different from Nielsen's in the algebraic methods he uses. Zimmermann gives an algebraic solution to the classification of any finite abelian group action on closed surfaces. His technique is to get every surjective homomorphism from the fundamental group of the surface into G in normal form and then differentiate between the normal forms. In this dissertation we classify fixed-point free finite abelian group actions on circle bundles. By results of Waldhausen every homeomorphism of M is isotopic to a fiber preserving homeomorphism; that is, it preserves the S1 factor of the bundle. This corresponds to the algebraic condition that any automorphism on the fundamental group of M preserves the center of the group. We use the same approach as that of Nielsen on surfaces. We give algorithms to bring every surjective homomorphism from the fundamental group of the bundle into the group G to normal form. From there we differentiate between the normal forms based on Nielsen's results. The results obtained are for circle bundles over surfaces of genus g greater than or equal to 2. A complete classification is given for the case that the circle bundle is a product bundle and G is a finite abelian group. We also obtain a complete classification of cyclic group actions and finite abelian group actions on circle bundles that are not product bundles.