Algebraic & Geometric Topology 7 (2007) 1081–1097

DOI: 10.2140/agt.2007.7.1081

arXiv: math.AT/0504152

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.

Keywords

immersions, bordism, cobordism, Herbert's formula

Mathematical Subject Classification

Primary: 57R42

Secondary: 55N22, 57R90

References

Publication

Received: 20 December 2006
Revised: 30 March 2007
Accepted: 5 April 2007
Published: 2 August 2007

Authors