Null k-cobordant links in \(S^ 3\) (Q810946)

A k-cobordism between links with m components in \(S^ 3\) is given by m compact connected oriented surfaces \(V_ i\), disjointly embedded in \(S^ 3\times I\), with prescribed boundary behaviour and such that certain elements of \(\pi_ 1(V_ i)\), when pushed into the complement, lie already in the k-th term of the lower central series of the fundamental group of \(S^ 3\times I-\cup V_ i\). In the present paper the resulting equivalence relation is compared to link homotopy. In particular, it is shown that if a link is null k-cobordant then all its \({\bar \mu}\)-invariants with length less than or equal to 2k vanish. This completes the proof of a conjecture of Cochran and Orr.