  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.
