Homology of coverings (Q762806)

The author is concerned with the following question: Does every closed, orientable, irreducible 3-manifold M with \(\pi_ 1(M)\) infinite have a finite sheeted cover \(\tilde M\) with infinite \(H_ 1(\tilde M) ?\) Such an \(\tilde M\) would be a Haken manifold, so the positive answer to this question would imply existence of finite sheeted Haken covers of M. A general, though somewhat awkward method for computing one-dimensional homology of finite covers of a finite CW-complex X from a presentation of \(\pi_ 1(X)\) was developed by Fox in the early fifties. The author shows how to simplify Fox's technique in the cases of regular abelian and irregular dihedral coverings. This simplified technique is combined with the author-reviewer theory of dual presentations of fundamental groups of 3-manifolds to yield for 3-manifolds with various types of symmetry some conditions which guarantee existence of finite covers with infinite \(H_ 1\). In particular, the author proves that if M (as above) has an involution \(\tau\) : \(M\to M\) and a Heegaard decomposition \(M=V_ 1\cup V_ 2\) such that deg \(\tau\) \(=-1\), \(H_ 1(M/\tau;{\mathbb{Z}}/2{\mathbb{Z}})\neq 0\) and \(\tau\) interchanges \(V_ 1\) and \(V_ 2\) then M has a finite cover with infinite \(H_ 1\).