# Bordism groups of immersions and classes represented by self-intersections

Eccles, Peter J and Grant, Mark (2006) Bordism groups of immersions and classes represented by self-intersections. [MIMS Preprint]

## Abstract

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 self-intersections and stable Hopf invariants and the fact that bordism of immersions gives a functor on the category of smooth manifolds and proper immersions.

Item Type: MIMS Preprint immersions, bordism, cobordism, Herbert's formula MSC 2010, the AMS's Mathematics Subject Classification > 55 Algebraic topologyMSC 2010, the AMS's Mathematics Subject Classification > 57 Manifolds and cell complexes Dr Peter J Eccles 20 Dec 2006 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/674