Hermitian forms on link modules
Simple links of odd dimensions \(\geq 3\) represnt the class of links admitting nice algebraic classification. The source of such classification might be thought to come from classifications of simple knots. So, concerning this paper, we may think that the theorem of \textit{H. F. Trotter} [Invent. Math. 20, 173-207 (1973; Zbl 0269.15009)] and \textit{C. Kearton} [Trans. Am. Math. Soc. 202, 141-160 (1975; Zbl 0305.57016)], which concerns simple knots and states that the Alexander module together with the Blanchfield form determine the knot uniquely, is generalized to simple links. Of course, an extensive study of algebraic properties of appropriate generalization of the Alexander module is needed and it occupies most of the paper. The obtained algebraic results combined with the results of the same author proved in his paper ``Stable homotopy and homology invariants of boundary links'' [Trans. Am. Math. Soc. (to appear)] enable him to prove the following classification theorem (Theorem 6.7 in the paper): Let \(q\geq 3\). The \(\Lambda\)-module \(H_ q(\tilde X)\) together with the pairing (Blanchfield form) \([, ]: H_ q\tilde X\times H_ q(\tilde X)\to \Gamma /\Lambda,\) provide a complete system of invariants for (q-1)-simple, (2q-1)-dimensional \({\mathcal F}\)-links. In other words, two (q-1)-simple (2q-1)-dimensional \({\mathcal F}\)-links are equivalent iff there exists an isomorphism between the corresponding modules \(H_ q(\tilde X)\), preserving the pairings. To explain the notation, let us mention that \(X=S^{2q+1}-T(\Sigma)\) is the complement of a tubular neighborhood T(\(\Sigma\)) of the link \(\Sigma\), \(\tilde X\to X\) the covering corresponding to the kernel of the splitting \({\mathcal S}: \pi_ 1(X,*)\to F_{\mu}\) \((=\) the free group in \(\mu\) generators, \(\mu\) \(=\) number of components of \(\Sigma\)); \(F_{\mu}\) acts on \(\tilde X\) as the group of covering translations; \(\Lambda ={\mathbb{Z}}[F_{\mu}]\), \(\Gamma ={\mathbb{Z}}[[x_ 1,...,x_{\mu}]]\) \(=\) the ring of formal power series in non-commuting variables \(x_ 1,...,x_{\mu}\); \(\Lambda\) is embedded in \(\Gamma\) via the Magnus embedding, and \(H_ q(\tilde X)=X_ q(\tilde X,{\mathbb{Z}})\) is the q- dimensional homology considered as left \(\Lambda ={\mathbb{Z}}[F_{\mu}]\)- module.
scientific article; zbMATH DE number 4213021
scientific article