## Abstract

Let GI denote the space of all generic immersions of a surface F into a 3-manifold M. Let q(H_{t}) denote the number mod 2 of quadruple points of a generic regular homotopy H_{t}:F → M. We are interested in defining an invariant Q : GI → ℤ/2 such that q(H_{t}) = Q(H_{0}) - Q(H_{1}) for any generic regular homotopy H_{t}:F → M. Such an invariant exists iff q = 0 for any closed generic regular homotopy (abbreviated CGRH). We prove that indeed q(H_{t}) = 0 for any CGRH H_{t} : F → ℝ^{3} where F is any system of surfaces. We prove the same for general 3-manifolds in place of ℝ^{3} under certain assumptions. We demonstrate the need for these assumptions with various counter-examples. We give an explicit formula for the invariant Q for embeddings of a system of tori in ℝ^{3}.

Original language | English |
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Pages (from-to) | 1069-1088 |

Number of pages | 20 |

Journal | Topology |

Volume | 39 |

Issue number | 5 |

DOIs | |

State | Published - Sep 2000 |

Externally published | Yes |

## Keywords

- Immersions
- Quadruple points
- Regular homotopy