In this thesis, we study the elliptic involution from the point of view of the bordered Heegaard Floer homology.

We show that the bordered Heegaard Floer invariant CFD^hat of a knot complement in S^3 is invariant under the elliptic involution on its boundary. As a computational result, we demonstrate how the bordered Heegaard Floer invariant CFDA^hat associated to the elliptic involution affects such CFD^hat complexes via A infinity tensor product. Globally, it "flips" the entire complex, while locally, it modifies the complex nicely in an arrow-wise fashion. The latter local description can be extended to quarter boundary Dehn twists as well.

Additionally, the Heegaard Floer homology of the associated open book decomposition to the elliptic involution is computed.

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 π_1-injective continuous map f: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 f lifts to M'? We approach this problem by reformulating it in terms of isometric actions of π_1(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 f:S → M, which can be used to derive cohomological criteria that guarantee the existence of a lift of f to a non-trivial connected finite-sheeted covering space of M.