Bordism groups of immersions and classes represented by self-intersections

Eccles, Peter J¹ ; Grant, Mark²

¹School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
²Department of Mathematical Sciences, Durham DH1 3LE, UK

Algebraic and Geometric Topology, Tome 7 (2007) no. 2, pp. 1081-1097

Résumé

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.

Détail

DOI : 10.2140/agt.2007.7.1081

Keywords: immersions, bordism, cobordism, Herbert's formula

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