### Abstract

We present a formula describing the action of a generalised Steenrod operation of Z(2)-type [14] on the cohomology class represented by a proper self-transverse immersion f: Ma dagger not signX. Our formula depends only on the Umkehr map, the characteristic classes of the normal bundle, and the class represented by the double point immersion of f. This generalises a classical result of R. Thom [13]: If alpha is an element of H (k) (X; Z(2)) is the ordinary cohomology class represented by f: Ma dagger not signX, then Sq (i) (alpha) = f(*) w (i) (nu(f) ).

Original language | English |
---|---|

Pages (from-to) | 272-281 |

Number of pages | 10 |

Journal | Acta Mathematica Hungarica |

Volume | 137 |

Issue number | 4 |

Early online date | 3 Jan 2012 |

DOIs | |

Publication status | Published - Dec 2012 |

### Keywords

- immersion
- self-intersection
- Steenrod-tom Dieck operation
- Steenrod square
- geometric cobordism

### Cite this

*Acta Mathematica Hungarica*,

*137*(4), 272-281. https://doi.org/10.1007/s10474-011-0189-9

**Self-intersections of immersions and Steenrod operations.** / Eccles, Peter J.; Grant, Mark.

Research output: Contribution to journal › Article

*Acta Mathematica Hungarica*, vol. 137, no. 4, pp. 272-281. https://doi.org/10.1007/s10474-011-0189-9

}

TY - JOUR

T1 - Self-intersections of immersions and Steenrod operations

AU - Eccles, Peter J.

AU - Grant, Mark

PY - 2012/12

Y1 - 2012/12

N2 - We present a formula describing the action of a generalised Steenrod operation of Z(2)-type [14] on the cohomology class represented by a proper self-transverse immersion f: Ma dagger not signX. Our formula depends only on the Umkehr map, the characteristic classes of the normal bundle, and the class represented by the double point immersion of f. This generalises a classical result of R. Thom [13]: If alpha is an element of H (k) (X; Z(2)) is the ordinary cohomology class represented by f: Ma dagger not signX, then Sq (i) (alpha) = f(*) w (i) (nu(f) ).

AB - We present a formula describing the action of a generalised Steenrod operation of Z(2)-type [14] on the cohomology class represented by a proper self-transverse immersion f: Ma dagger not signX. Our formula depends only on the Umkehr map, the characteristic classes of the normal bundle, and the class represented by the double point immersion of f. This generalises a classical result of R. Thom [13]: If alpha is an element of H (k) (X; Z(2)) is the ordinary cohomology class represented by f: Ma dagger not signX, then Sq (i) (alpha) = f(*) w (i) (nu(f) ).

KW - immersion

KW - self-intersection

KW - Steenrod-tom Dieck operation

KW - Steenrod square

KW - geometric cobordism

U2 - 10.1007/s10474-011-0189-9

DO - 10.1007/s10474-011-0189-9

M3 - Article

VL - 137

SP - 272

EP - 281

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 4

ER -