1999/09/03 by Tahl Nowik, Nowik, Tahl
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #math.GT
paper · pdf · doi:10.48550/arxiv.math/9909020
24 pages, 10 figures
arxiv created 1999/09/03 · openalex publication_date 1999/09/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.