### 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.

Original language | English |
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Pages (from-to) | 1081-1097 |

Number of pages | 17 |

Journal | Algebraic & Geometric Topology |

Volume | 7 |

DOIs | |

Publication status | Published - 2007 |

### Keywords

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

### Cite this

*Algebraic & Geometric Topology*,

*7*, 1081-1097. https://doi.org/10.2140/agt.2007.7.1081