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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

arXiv: math.AT/0504152


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.

immersions, bordism, cobordism, Herbert's formula
Mathematical Subject Classification 2000
Primary: 57R42
Secondary: 57R90, 55N22
Received: 20 December 2006
Revised: 30 March 2007
Accepted: 5 April 2007
Published: 2 August 2007
Peter J Eccles
School of Mathematics
University of Manchester
Oxford Road
Manchester M13 9PL
Mark Grant
Department of Mathematical Sciences
Durham DH1 3LE