Manchester Geometry Seminar 2004/2005

For an immmersion f: M --> N in general position, the 'r-fold self intersections' of f give immersed manifolds in M and N. These immersions represent classes in the (co)homology of the manifolds, and in his 1980 thesis R.J. Herbert gives a formula relating these classes, involving the Euler class of the normal bundle of f. We attempt to generalise and elucidate this result. To do so we study cobordism of immersions as a contravariant functor on the category of smooth manifolds and immersions. In this setting constructions such as induced maps and products become geometrically transparent. This functor has a homotopy interpretation as the stable homotopy of Thom complexes.