Three dualities on the integral homology of infinite cyclic coverings of manifolds
Let M be a compact oriented manifold and \(\gamma\) a homomorphism from \(\pi_ 1(M)\) to \({\mathbb{Z}}\), and let \(H_*(M;\gamma)\) denote the integral homology of the infinite cyclic cover of M determined by \(\gamma\). These homology groups are finitely generated \(\Lambda\)-modules, where \(\Lambda ={\mathbb{Z}}[t,t^{-1}]\) is the group ring of the covering group. This paper establishes duality theorems generalizing those of Blanchfield for the maximal \(\Lambda\)-torsion free and pure torsion subquotients of \(H_*(M;\gamma)\) and that of Farber and Levine for the maximal finite submodule. (A finitely generated \(\Lambda\)-torsion module is pure if it has no nontrivial finite submodule.) The key ideas used are a Universal Coefficient exact sequence (established directly, but derivable from a standard spectral sequence), equivariant Poincaré duality (here called Reidemeister duality), and the good homological properties of the ring \(\Lambda\).
integral homology of the infinite cyclic cover
scientific article; zbMATH DE number 4069842