Three dualities on the integral homology of infinite cyclic coverings of manifolds
type: Item
instance of: scholarly article
title: Three dualities on the integral homology of infinite cyclic coverings of manifolds
zbMATH Open document ID: 0655.57013
author: Akio Kawauchi
published in: Osaka Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: 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 \(\Lambda\).
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 57M10
zbMATH DE Number: 4069842
zbMATH Keywords: Blanchfield duality
zbMATH Keywords: Farber-Levine duality
zbMATH Keywords: integral homology of the infinite cyclic cover
zbMATH Keywords: pure torsion subquotients
zbMATH Keywords: maximal finite submodule
zbMATH Keywords: equivariant Poincaré duality
reviewed by: Jonathan A. Hillman
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4069842