Quadruple points of regular homotopies of surfaces in 3-manifolds

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Let GI denote the space of all generic immersions of a surface F into a 3-manifold M. Let q(Ht) denote the number mod 2 of quadruple points of a generic regular homotopy Ht:F → M. We are interested in defining an invariant Q : GI → ℤ/2 such that q(Ht) = Q(H0) - Q(H1) for any generic regular homotopy Ht:F → M. Such an invariant exists iff q = 0 for any closed generic regular homotopy (abbreviated CGRH). We prove that indeed q(Ht) = 0 for any CGRH Ht : F → ℝ3 where F is any system of surfaces. We prove the same for general 3-manifolds in place of ℝ3 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 ℝ3.

Original languageEnglish
Pages (from-to)1069-1088
Number of pages20
Issue number5
StatePublished - Sep 2000
Externally publishedYes


  • Immersions
  • Quadruple points
  • Regular homotopy


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