This paper deals with an
analysis of the first homology of a finite sheeted covering space of a complex and
gives applications to some questions about 3-manifolds. Section 2 considers the
relation between the property that a 3-manifold be virtually Haken and the,
seemingly stronger, property that some finite sheeted cover has positive first betti
number. Section 3 gives a procedure for computing the homology of a finite cover in
terms of a presentation of the fundamental group of the base, and its action on the
fiber and includes generalizations of the Fox-Goeritz theorem for cyclic covers to
arbitrary abelian covers and to dihedral covers. Section 4 applies these theorems to
3-manifolds which have various types of symmetry and include some conditions which
guarantee finite covers with positive first betti number. The paper concludes with a
section of examples.
