Homology boundary links and the Andrews-Curtis conjecture.

*(English)*Zbl 0734.57015This paper is concerned with links of n-spheres in \(S^{n+2}\). The main result is: Each sublink of a homology boundary link is concordant to a fusion of a boundary link. The importance of this result comes from the fact that boundary links up to concordance are well-understood. Moreover, recent results of J. P. Levine, W. Mio and K. E. Orr [to appear] and J. P. Levine [Invent. Math. 96, 571-592 (1989; Zbl 0692.57010)] show that sublinks of homology boundary links are (up to concordance) characterized by the vanishing of homotopy obstructions. The result above states that the gap between the two classes is “bridged” by a natural band-attaching construction. The authors prove that dropping the concordance condition in the main result is naturally related to the Andrews Curtis Conjecture: If L is a homology boundary link with free group, then L is fusion of a boundary link if and only if its pattern is invertible. Here the pattern is a precise obstruction for a homology boundary link to be a boundary link. A weakening of the AC-conjecture is that all patterns are invertible.

Reviewer: U.Kaiser (Siegen)

##### MSC:

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

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |