Subgroups of the Mapping Class Group and Quadruple Points of Regular Homotopies

Details Subgroups of the Mapping Class Group and Quadruple Points of Regular Homotopies Subgroups of the Mapping Class Group and Quadruple Points of Regular Homotopies

1999-09-03

arxiv.org/abs/math/9909020v1

Mathematics

Geometric Topology

24 pages, 10 figures

Scientific paper

Let F be a closed orientable surface. If i,i':F \to R^3 are two regularly homotopic generic immersions, then it has been shown in [N] that all generic regular homotopies between i and i' have the same number mod 2 of quadruple points. We denote this number by Q(i,i') \in Z/2. We show that for any generic immersion i:F\to R^3 and any diffeomorphism h:F\to F such that i and i\circ h are regularly homotopic, Q(i,i\circ h) = (rank(h_*-Id) + (n+1)e(h)) mod 2, where h_* is the map induced by h on H_1(F,Z/2), n is the genus of F and e(h) is 0 or 1 according to whether h is orientation preserving or reversing, respectively.

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