Applications of noncommutative intersection forms to linking

Abstract

We construct a link homotopy invariant for three-component spherical link maps which is a generalisation of the Kirk invariant for two-component spherical link maps. We then construct an invariant for three-component annular link maps and establish that there is a relationship between the three-component annular link map invariant and the three-component link map invariant. We show the link map invariant can detect non-trivial three-component link maps which become trivial up to link homotopy when a component is removed. We establish that there exist link maps where each component has the same image in but are not link homotopic, unlike in the two-component case. Using the link map invariant we construct an invariant which is analogous to Milnor's triple linking number, and show that they can be used to distinguish different link maps. We provide a method for calculating our invariant for an infinite family of three-component annular link maps and deduce a computation for link maps and detect infinitely many three-component link maps which have vanishing Kirk invariants.

Next we prove that the Blanchfield form on a closed, orientable, connected three-manifold can be computed in terms of the intersection form of a four-manifold which it bounds. We aim to do this using the weakest assumptions possible, as similar results in the literature have been shown but proofs are often imprecise or make more assumptions than are needed.