Periodic ends and knot concordance
type: Item
instance of: scholarly article
title: Periodic ends and knot concordance
zbMATH Open document ID: 0687.57004
DOI: 10.1016/0166-8641(89)90051-5
author: Robert E. Gompf
published in: Topology and its Applications
publication date: 1989-01-01T00:00:00Z
review text: Two knots \(K_ 0\) and \(K_ 1\) in \(S^ 3\) are smoothly concordant if there is a smooth annulus embedded in \(S^ 3\times I\) with boundary \(K_ 0\times \{0\}\cup K_ 1\times \{1\}\). The set \({\mathcal K}_{DIFF}\) of smooth concordance classes of knots is an abelian group under connected sum. Elements of the identity class are called smoothly slice. Similarly, the topological concordance group \({\mathcal K}_{TOP}\) is obtained by using flat topological annuli in place of smooth ones. Inclusion gives a canonical epimorphism i: \({\mathcal K}_{DIFF}\to {\mathcal K}_{TOP}\). The main result of the paper is that the kernel of i is larger than Z, even modulo torsion. The proof uses the Taubes theorem on nonexistence of certain definite 4-manifolds with periodic ends. A corollary of the proof is that there are uncountably many diffeomorphism types of Casson handles. A specific collection of doubled knots is exhibited, no two members of which are smoothly concordant.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 58J90
Mathematics Subject Classification ID: 57R10
zbMATH DE Number: 4126320
zbMATH Keywords: gauge theory
zbMATH Keywords: Whitehead double
zbMATH Keywords: topological knot concordance
zbMATH Keywords: topologically slice knots
zbMATH Keywords: smoothly slice knots
zbMATH Keywords: knots
zbMATH Keywords: smoothly concordant
zbMATH Keywords: smooth concordance classes of knots
zbMATH Keywords: definite 4-manifolds with periodic ends
zbMATH Keywords: uncountably many diffeomorphism types of Casson handles
zbMATH Keywords: doubled knots
reviewed by: Józef H. Przytycki
MaRDI profile type: Publication
full work available at URL: 0166-8641(89)90051-5
OpenAlex ID: W2049120537
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description: scientific article; zbMATH DE number 4126320