The linking number of singular maps (Q1081896)

The set of multiple points of a given multiplicity of an immersion of a closed manifold into a Euclidean space realizes the homology class dual to the appropriate power of the top dimensional normal Stiefel-Whitney class. This statement remains true for singular maps as well if one considers only double points. In the paper we consider the case of triple points of singular maps \(M^{2k}\to {\mathbb{R}}^{3k}\). In this case the corresponding formula (for immersions) does not hold anymore. The difference of the two sides can be interpreted geometrically as the ''linking number of the map''. The latter is defined as the linking number of the image of the map with the submanifold of singular points pushed off from the image set in a certain, well-defined direction, namely in the direction of the outward normal vectorfield in the image set of the double points.