Not All Links Are Concordant to Boundary Links

A link is a smooth, oriented submanifold L = {Kx, . . . , Km} of S which is the ordered disjoint union of m manifolds each piecewise-linearly homeomorphic to the «-sphere (if m = 1, L is called a knot). Knots and links play an essential role in the classification of manifolds and, in this regard, perhaps the most important equivalence relation on links is that of link concordance. LQ and L{ are concordant if there is a smooth, oriented submanifold C = {Cx, . . . , Cm} of S x [0,1] which meets the boundary transversely in dC, is piecewise-linearly homeomorphic to L0 x [0, 1] and meets S n+2 x {/} in L. for / = 0, 1. The particular situation which led to the introduction of this equivalence relation and which indicates its importance is as follows. If S is an immersed 2-disk or 2-sphere in a 4-manifold X, x0 is a singular value and B is a small 4-ball neighborhood of x 0 , then S n B is a link in *S. If L were concordant to a link whose components bound disjoint 2-disks in S (the latter is called a trivial link) then the singularity at x0 could be removed. Thus the fundamental problem is to classify (for fixed m, n) the set of concordance classes. In the mid-1960s M. Kervaire and J. Levine gave an algebraic classification of the high-dimensional (n > 1) knot concordance groups [L2]. For even n these are the trivial group and for odd n they are infinitely generated. In a sequence of papers S. Cappell