Bordism groups of immersions and classes represented by self-intersections

Peter J. Eccles*, Mark Grant

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1081-1097
Number of pages17
JournalAlgebraic & Geometric Topology
Publication statusPublished - 2007


  • multiple points
  • invariants
  • immersions
  • bordism
  • cobordism
  • Herbert's formula

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