Bordism groups of immersions and classes represented by self-intersections

A well known formula of R J Herbert's relates the various homology classes represented by the self intersection immersions of a self transverse immersion. We prove a geometrical version of Herbert's formula by considering the self intersection immersions of a self transverse immersion up to bordism. This clarifies the geometry lying behind Herbert's formula and leads to a homotopy commutative diagram of Thom complexes. It enables us to generalise the formula to other homology theories. The proof is based on Herbert's but uses the relationship between selfintersections and stable Hopf invariants and the fact that bordism of immersions gives a functor on the category of smooth manifolds and proper immersions. © 2007 Mathematical Sciences Publishers.