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

الأصل · EN

Let F be a closed orientable surface. If i,i':F → R³ 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') ∈ Z/2. We show that for any generic immersion i:F→ R³ and any diffeomorphism h:F→ F such that i and i∘ h are regularly homotopic, Q(i,i∘ h) = (rank(h*-Id) + (n+1)e(h)) mod 2, where h* is the map induced by h on H₁(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.