The peripheral subground and the second homology of the group of a knotted torus in \(S^ 4\)
type: Item
instance of: scholarly article
title: The peripheral subground and the second homology of the group of a knotted torus in \(S^ 4\)
zbMATH Open document ID: 0871.57025
author: Taizo Kanenobu
author: Ken-ichiro Kazama
published in: Osaka Journal of Mathematics
publication date: 1997-09-30T00:00:00Z
review text: Let \(F\) be an embedded torus in \(S^4\), and let \(\pi F=\pi_1(S^4-F)\). Let \(\tau F\) be the the image of \(\pi_1(F)\cong Z^2\) in \(\pi F\). If \(A\) and \(B\) are abelian groups write \(A\leq B\) if \(A\) is a quotient of \(B\). Then \(\tau F\leq Z^2\) and \textit{R. A. Litherland} [Q. J. Math., Oxf. II. Ser. 32, 425-434 (1981; Zbl 0506.57013)] showed that \(H_2(\pi F;Z)\leq\tau F\), and that any quotient of \(Z^2\) is realizable as \(H_2(\pi F;Z)\) for some torus \(F\) with \(\tau F\cong Z^2\). This paper considers the question of realizing pairs \(A\), \(B\) such that \(A\leq B\leq Z^2\), and shows that this is possible provided that either \(B\) is torsion free, or \(B\) has rank one and \(A\) is finite, or \(B\) is cyclic. (The remaining cases are undecided).
Mathematics Subject Classification ID: 57Q45
zbMATH DE Number: 755429
zbMATH Keywords: peripheral subgroup
zbMATH Keywords: knotted surface
zbMATH Keywords: group homology
zbMATH Keywords: satellite knot
zbMATH Keywords: embedded torus
reviewed by: Jonathan A. Hillman
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 755429