Vol. 112, No. 1, 1984

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Homology of coverings

John Paul Hempel

Vol. 112 (1984), No. 1, 83–113

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.

Mathematical Subject Classification 2000
Primary: 57M05
Secondary: 57N10
Received: 9 September 1981
Published: 1 May 1984
John Paul Hempel