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Link concordance and algebraic closure of groups. (English) Zbl 0692.57009

For any subgroup \(G\subset \hat F\) \((=\) nilpotent completion of the free group F of rank r) the author defines the algebraic closure to be the subgroup \(\bar G\subset \hat F\) of those elements which are part of a solution to an arbitrary finite system of equations \(x_ i=w_ i\in G*F(x_ 1,...,x_ n)\), where each \(w_ i\) is a product of conjugates of elements of G. The author shows the relevance of this notions for the study of link concordance:
1) K. Orr defined a link concordance invariant \(\theta_{\omega}(L)\in \pi_{n+2}(K_{\omega})\), where \(K_{\omega}\) is the mapping cone of K(F,1)\(\to K(\hat F,1)\) [Comment. Math. Helv. 62, 542-560 (1987; Zbl 0637.57016)]. The author defines a lift \(\theta_{\infty}(L)\in \pi_{n+2}(K_{\infty})\) of this invariant, where \(K_{\infty}\) is defined as K by replacing \(\hat F\) by \(\bar F.\) Then he proves (by an exciting journey through algebra and surgery theory) that \(\pi_{n+2}(K_{\infty})\) can be identified (via Pontryagin-Thom-construction) with (based) \(\omega\)-concordance classes of (based) \(\omega\)-links in \(S^{n+2}\). Here an \(\omega\)-link is a codimension 2 manifold satisfying an additional condition on “longitudes”. This is the natural generalization of higher-dimensional spherical links and links in \(S^ 3\) with vanishing \({\bar \mu}\)- invariants. Unfortunately the groups \(\pi_{n+2}(K_{\infty})\) are not known for \(n>1.\)
2) If EW is the Vogel localization of a wedge of r circles, then \(\bar F\) is naturally isomorphic to \(\pi_ 1(EW)/\pi_ 1(EW)_{\omega}\). The Vogel localization was used by J.-Y. Le Dimet in an exact sequence for disk link concordance [Mem. Soc. Math. France, Nouv. Sér. 32 (1988; Zbl 0666.57015)]. The author proves that \(\theta_{\infty}\) vanishes on the class of finite E-links, which contains the class of SHB-links (sublinks of homology boundary links). It has recently been proved by Levine/Mio/Orr that the vanishing of Le Dimet’s homotopy obstruction for a high-dimensional disk link L implies that L is concordant to an SHB- link. But the relation between Le Dimet’s invariant and \(\theta_{\infty}\) is not clear to the reviewer. The author finally proves that a finite E-link in \(S^ 3\) (with E-group G) is concordant to an SHB-link if and only if an obstruction in \(H_ 3G\) vanishes. In fact, modulo the Whitehead conjecture, each finite E-link is concordant to an SHB-link.
Reviewer: U.Kaiser

MSC:

57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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