Minimal Seifert manifolds and the knot finiteness theorem
type: Item
instance of: scholarly article
title: Minimal Seifert manifolds and the knot finiteness theorem
zbMATH Open document ID: 0692.57011
DOI: 10.1007/BF02765892
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: It is a well known consequence of obstruction theory and transversality that every n-knot (oriented locally flat submanifold of \(S^{n+2}\) homeomorphic to \(S^ n)\) bounds an embedded ``Seifert'' hypersurface. The infinite cyclic covering space \(\tilde X\) of the knot complement X may be constructed by splitting X along any such Seifert hypersurface V to obtain a manifold Y whose boundary contains two copies \(V_-\) and \(V_+\) of V, and identifying in \(Y\times {\mathbb{Z}}\) each \(V_+\times \{n\}\) with \(V_-\times \{n+1\}\), for all n in \({\mathbb{Z}}\). This paper studies Seifert hypersurfaces V which are minimal in the sense that the natural inclusions of V into \(\tilde X\) induce monomorphisms on homology. The homology groups of \(\tilde X\) are modules over the group ring \(\Lambda ={\mathbb{Z}}[{\mathbb{Z}}]={\mathbb{Z}}[t,t^{-1}]\). A key idea of this paper is to relate V to \(\tilde X\) via the subspace which is the image of \(Y\neq N\) in \(\tilde X\) (where N is the set of natural numbers); the homology groups of this subspace are modules over the semigroup ring \(\Lambda_+={\mathbb{Z}}[N]={\mathbb{Z}}[t]\). Two more rings play a role in the argument. As X is a homology circle the element 1-t of \(\Lambda\) acts invertibly on the homology groups of \(\tilde X,\) which are thus modules over \(L=\Lambda [1-t)^{-1}]\). The Seifert pairing determines an endomorphism z of the stable homotopy type of V and so the homology groups of V are modules over \(P={\mathbb{Z}}[z]\). If P is imbedded in L via \(z=(1-t)^{-1}\) the natural inclusions of V into \(\tilde X\) induce P- isomorphisms on homology. The main result is then Theorem 2.3, in which it is shown that if K is an n-knot with \(n\geq 4\) and knot group \(\pi\) \(K\cong {\mathbb{Z}}\) then there is a 1-connected Seifert hypersurface for K realizing (up to isomorphism) any given family of P-lattices in the homology of \(\tilde X;\) if \(n=2q-1\) we must require also that the middle dimensional lattice b…
reviewed by: Jonathan A. Hillman
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 4133130
zbMATH Keywords: Blanchfield pairing
zbMATH Keywords: n-knot
zbMATH Keywords: Seifert hypersurface
zbMATH Keywords: Seifert pairing
zbMATH Keywords: 1- connected Seifert hypersurface
zbMATH Keywords: middle dimensional lattice
zbMATH Keywords: intersection form on the Seifert hypersurface
zbMATH Keywords: odd-dimensional knots
zbMATH Keywords: stable n-knot
zbMATH Keywords: Alexander modules
author: Michael Farber
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4133130