### 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 |
---|---|

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

**Bordism groups of immersions and classes represented by self-intersections.** / Eccles, Peter J.; Grant, Mark.

Research output: Contribution to journal › Article

*Algebraic & Geometric Topology*, vol. 7, pp. 1081-1097. https://doi.org/10.2140/agt.2007.7.1081

}

TY - JOUR

T1 - Bordism groups of immersions and classes represented by self-intersections

AU - Eccles, Peter J.

AU - Grant, Mark

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

KW - multiple points

KW - invariants

KW - immersions

KW - bordism

KW - cobordism

KW - Herbert's formula

U2 - 10.2140/agt.2007.7.1081

DO - 10.2140/agt.2007.7.1081

M3 - Article

VL - 7

SP - 1081

EP - 1097

JO - Algebraic & Geometric Topology

JF - Algebraic & Geometric Topology

SN - 1472-2747

ER -