n-Quasi-isotopy: I. Questions of nilpotence

Sergey A. Melikhov, Dusan Repovs

Published 2001-03-18, updated 2005-08-19Version 11

It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for some (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C^0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-isotopy, k=1,2,...; all of them are weaker than isotopy (in the sense of Milnor). We prove that every link can be cancelled up to peripheral structure preserving isomorphism of any quotient of the fundamental group, functorially invariant under k-quasi-isotopy; functoriality means that the isomorphism between the quotients for links related by an allowable crossing change fits in the commutative diagram with the fundamental group of the complement to the intermediate singular link. The proof invokes Baer's theorem on the join of subnormal locally nilpotent subgroups. On the other hand, the integral generalized (lk\ne 0) Sato-Levine invariant \tilde\beta is invariant under 1-quasi-isotopy, but is not determined by any quotient of the fundamental group (endowed with the peripheral structure), functorially invariant under 1-quasi-isotopy - in contrast to Waldhausen's theorem. As a byproduct, we use \tilde\beta to determine the image of the Kirk-Koschorke invariant \tilde\sigma of fibered link maps.