## Abstract

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R^{3}, with values in any Abelian group. We show they are all functions of a universal order 1 invariant which we construct as T ⊕ P ⊕ Q, where T is a Z valued invariant reflecting the number of triple points of the immersion, and P, Q are Z / 2 valued invariants characterized by the property that for any regularly homotopic immersions i, j : F → R^{3}, P (i) - P (j) ∈ Z / 2 (respectively, Q (i) - Q (j) ∈ Z / 2) is the number mod 2 of tangency points (respectively, quadruple points) occurring in any generic regular homotopy between i and j. For immersion i : F → R^{3} and diffeomorphism h : F → F such that i and i ○ h are regularly homotopic we show:P (i ○ h) - P (i) = Q (i ○ h) - Q (i) = (rank (h_{*} - Id) + ε (det h_{* *})) mod 2 where h_{*} is the map induced by h on H_{1} (F ; Z / 2), h_{* *} is the map induced by h on H_{1} (F ; Q), and for 0 ≠ r ∈ Q, ε (r) ∈ Z / 2 is 0 or 1 according to whether r is positive or negative, respectively.

Original language | English |
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Pages (from-to) | 1881-1893 |

Number of pages | 13 |

Journal | Topology and its Applications |

Volume | 154 |

Issue number | 9 |

DOIs | |

State | Published - 1 May 2007 |

### Bibliographical note

Funding Information:E-mail address: tahl@math.biu.ac.il. URL: http://www.math.biu.ac.il/~tahl. 1 Partially supported by the Minerva Foundation.

## Keywords

- Finite order invariants
- Immersions of surfaces