Symmetric surgery and boundary link maps

Abstract

In order to seperate 3-dimensional linking and knotting phenomena, John Milnor introduced the notion of a link homotopy [14]. He allowed selfintersections but did not allow different components to cross during a link homotopy. It is clear that any knot is link homotopically trivial but one of the most surprising and unintuitive (see the left hand side of Fig. 1) results of Milnor was that arbitrarily many parallel copies of a knot form a homotopically trivial link. In fact, one knows that any boundary link has vanishing μ-invariants and thus it is homotopically trivial by the main result of Milnor. By definition, in a boundary link the components bound disjointly embedded Seifert surfaces. For example, any knot has a Seifert surface and is thus a boundary link. Similarly, parallel copies of a knot bound (disjoint) parallel copies of this Seifert surface. The notion of a link homotopy makes clearly sense for links in any dimension. In fact, the correct category to work in seems to be the following: A link map is a continuous map such that connected components in the source are mapped disjointly into the target. A link homotopy between two link maps is then an ordinary homotopy through link maps. In this context, a boundary link map L : qi Mi → N is a link map which is the boundary of a link map qi Wi → N of oriented manifolds Wi, the “Seifert surfaces” for L.