Finite-Sheeted Covering Spaces and Solenoids over 3-manifolds

Abstract

This thesis develops techniques for studying towers of finite-sheeted covering spaces of 3-manifolds. The central question we seek to address is the following: given a π₁-injective continuous map ƒ : S → M of a 2-manifold S into a 3-manifold M, when does there exist a non-trivial connected finite-sheeted covering space M' of M such that ƒ lifts to M'? We approach this problem by reformulating it in terms of isometric actions of π₁(M) on compact metric spaces. We then study regular solenoids over M, which give natural examples of compact metric spaces with isometric π ₁(M)-actions. We conclude by introducing a construction that we call the mapping solenoid of a map ƒ : S → M, which can be used to derive cohomological criteria that guarantee the existence of a lift of ƒ to a non-trivial connected finite-sheeted covering space of M.