Degree-1 maps into lens spaces and free cyclic actions on homology 3- spheres
type: Item
instance of: scholarly article
title: Degree-1 maps into lens spaces and free cyclic actions on homology 3- spheres
zbMATH Open document ID: 0713.57008
DOI: 10.1016/0166-8641(90)90057-9
author: D. Kharzeev
published in: Topology and its Applications
publication date: 1990-01-01T00:00:00Z
review text: The paper deals with the existence of degree-1 maps f: \(M\to L_{n,m}\) from a closed orientable 3-manifold M to some lens space \(L_{n,m}\). The main result of the paper asserts that if \(H_ 1(M)={\mathbb{Z}}_ n\), then such a map f exists and the lens space \(L_{n,m}\) is determined uniquely (up to homotopy equivalence) by the mod n intersection number \(<\beta (x),x>=\pm mk^ 2\), where x is a generator of \(H_ 2(M,{\mathbb{Z}}_ n)={\mathbb{Z}}_ n\) and \(\beta: H_ 2(M,{\mathbb{Z}}_ n)\to H_ 1(M,{\mathbb{Z}}_ n)\) is the Bockstein homomorphism defined by the exact sequence \(0\to {\mathbb{Z}}_ n\to {\mathbb{Z}}_{n^ 2}\to {\mathbb{Z}}_ n\to 0\). As a corollary, the authors show that if \(\tilde M\) is a homology 3- sphere and p: \(\tilde M\to M\) is a regular covering with \({\mathbb{Z}}_ n\) as its group of covering transformations, then p is induced from the universal covering \(S^ 3\to L_{n,m}\) by a degree-1 map f: \(M\to L_{n,m}\).
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 57M99
zbMATH DE Number: 4173917
zbMATH Keywords: degree-1 maps from a closed orientable 3-manifold to some lens space
zbMATH Keywords: homology 3-sphere
zbMATH Keywords: regular covering
reviewed by: Krzysztof Pawałowski
MaRDI profile type: Publication
full work available at URL: 0166-8641(90)90057-9
OpenAlex ID: W2009356642
cites work: Q5676841
cites work: Construction of Group Actions of Four-Manifolds
cites work: Q4114528
cites work: Q3125202
cites work: On regular coverings of 3-manifolds by homology 3-spheres
cites work: Q4768351
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description: scientific article; zbMATH DE number 4173917