Minimal Seifert manifolds and the knot finiteness theorem
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 be self dual with respect to the Milnor form \(H_ q(\tilde X)\otimes H_ q(\tilde X)\to {\mathbb{Q}}\), and the corresponding isomorphism may be chosen to be an isometry with respect to the intersection form on the Seifert hypersurface. This result is a substantial generalization of earlier work of Levine, who showed that if \(\tilde X\) is r-connected then the knot has an r-connected Seifert hypersurface. The theorem of Kearton and Trotter that simple odd-dimensional knots with isometric Blanchfield or Milnor forms are isotopic follows easily from this work. Another more substantial application also given here is a new finiteness theorem: a stable n-knot is determined up to finite ambiguity by its set of Alexander modules. (The author had earlier obtained this result for fibred knots; the first such result apart from the classification of simple odd-dimensional knots was due to Hausmann, in the case when there is just one nonzero Alexander module below the middle dimension).
intersection form on the Seifert hypersurface
scientific article; zbMATH DE number 4133130
scientific article