Subgroups of the Mapping Class Group and Quadruple Points of Regular Homotopies
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Publication:6501468
arXivmath/9909020MaRDI QIDQ6501468
Author name not available (Why is that?)
Abstract: Let F be a closed orientable surface. If i,i':F o 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 o R^3 and any diffeomorphism h:F o F such that i and icirc h are regularly homotopic, Q(i,icirc 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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