Bordism groups of immersions and classes represented by self-intersections

Eccles, Peter J; Grant, Mark

doi:10.2140/agt.2007.7.1081

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Bordism groups of immersions and classes represented by self-intersections

Peter J Eccles and Mark Grant

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 2000

Primary: 57R42

Secondary: 57R90, 55N22

References

Publication

Received: 20 December 2006

Revised: 30 March 2007

Accepted: 5 April 2007

Published: 2 August 2007

Authors

Peter J Eccles
School of Mathematics University of Manchester Oxford Road Manchester M13 9PL UK
http://www.maths.manchester.ac.uk/~peter/
Mark Grant
Department of Mathematical Sciences Durham DH1 3LE UK
http://maths.dur.ac.uk/~dma1mg/

Volume 7, issue 2 (2007)