Narrow Results

Given a suitable link map f into a manifold M, we constructed, in a previous publication, link homotopy invariants κ(f) and μ(f). In the present paper we study the case M=Sⁿ×ℝ^m-n in detail. Here μ(f) turns out to be the starting term of a whole sequence μ^(s)(f), s=0, 1,…, of higher μ-invariants which together capture all the information contained in κ(f). We discuss the geometric significance of these new invariants. In several instances we obtain complete classification results. A central ingredient of our approach is the homotopy theory of wedges of spheres.

We fill a gap in the proof that the proposed link homotopy invariant $\omega$ of Li is well defined. It is also shown that if the homotopy invariant $\tau$ of Schneiderman–Teichner is to be adapted to a link homotopy invariant of link maps, the result coincides with $\omega$.

It is an open problem whether Kirk’s $\sigma$-invariant is the complete obstruction to a link map $f:S_{+}^2\cup S_{-}^2\to S^4$ being link homotopic to the trivial link. The link homotopy invariant associates to such a link map $f$ a pair $\sigma (f)=({\sigma }_{+}(f),{\sigma }_{-}(f))$, and we write $\sigma =({\sigma }_{+},{\sigma }_{-})$. With the objective of constructing counterexamples, Li proposed a link homotopy invariant $\omega =({\omega }_{+},{\omega }_{-})$ such that ${\omega }_{\pm }$ is defined on the kernel of ${\sigma }_{\pm }$ and which also obstructs link null-homotopy. We show that, when defined, the invariant ${\omega }_{\pm }$ is determined by ${\sigma }_{\mp }$, and is strictly weaker. In particular, this implies that if a link map $f$ has $\sigma (f)=(0,0)$, then after a link homotopy the self-intersections of $f(S_{+}^2)$ may be equipped with framed, immersed Whitney disks in $S^4\setminus f(S_{-}^2)$ whose interiors are disjoint from $f(S_{+}^2)$.