n-QUASI-ISOTOPY: I. QUESTIONS OF NILPOTENCE

It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link up to link homotopy can be cancelled in a (componentwise) connected sum with another link. In this paper we address the question whether some form of the ‘complexity accumulation’ property of knots holds for (piecewiselinear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-isotopy, k ∈ N; all of them are weaker than isotopy (in the sense of Milnor). We prove that no accumulation of complexity up to k-quasi-isotopy occurs from the viewpoint of any quotient of the fundamental group, functorially invariant under kquasi-isotopy (functoriality means that the isomorphism between the quotients for links related by an allowable crossing change fits in the commutative diagram with the inclusion induced homomorphisms from the fundamental group of the complement to the thickened intermediate singular link). On the other hand, the integral generalized (lk 6= 0) Sato–Levine invariant 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 a theorem by Waldhausen.