Algebraic & Geometric Topology

Bordism groups of immersions and classes represented by self-intersections

Peter J Eccles and Mark Grant

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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.

Article information

Algebr. Geom. Topol., Volume 7, Number 2 (2007), 1081-1097.

Received: 20 December 2006
Revised: 30 March 2007
Accepted: 5 April 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R42: Immersions
Secondary: 57R90: Other types of cobordism [See also 55N22] 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90]

immersions bordism cobordism Herbert's formula


Eccles, Peter J; Grant, Mark. Bordism groups of immersions and classes represented by self-intersections. Algebr. Geom. Topol. 7 (2007), no. 2, 1081--1097. doi:10.2140/agt.2007.7.1081.

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