Link concordance and algebraic closure. II
This is the sequel to Part I reviewed above (see Zbl 0692.57009). The author defines a group G to be algebraically closed (AC) if each 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, has a unique solution in G. For each group G he proves the existence of an algebraic closure \(G\to \hat G\) satisfying the universal property (in the usual sense) with respect to homomorphisms into AC-groups. Then he establishes the following ``localization'' properties: 1) \(G\to \hat G\) is normally surjective and homologically 2-connected. 2) For any normally surjective homomorphism \(\phi\) : \(G\to H\) (G finitely generated, H finitely presented), the induced homomorphism \(G\to \hat H\) is homologically 2- connected. The algebraic closure is the algebraic localization functor corresponding to the (topological) Vogel localization, in fact: if X is a finite complex with fundamental group G, then \(\hat G\) is the fundamental group of the Vogel localization of X. In the second part the author applies the concept of algebraic closure to link theory in \(S^ 3\). He defines the notion of \(\hat F-\)link (satisfying a condition ``ostensibly'' stronger than vanishing \({\bar \mu}\)-invariants; ``..'' is related to the canonical map \(\hat F\to \tilde F\) \(=\) nilpotent completion of the free group F). For those links a (based) concordance invariant \(\theta (L)\in H_ 3(\hat F)\) is defined. The author proves that for an \(\hat F-\)link L, \(\theta (L)=0\) if and only if L is concordant to a sublink of a homology boundary link. Moreover, each element in \(H_ 3\hat F\) is realized as \(\theta\) (L) for some \(\hat F-\)link L. In an appendix the author mentions several open problems relating interesting questions concerning the algebraic closure to important questions in link concordance.
fundamental group of the Vogel localization
scientific article; zbMATH DE number 4133129
scientific article