Automorphisms and Embeddings of Surfaces and Quadruple Points of Regular Homotopies

Abstract

Let F be a closed surface. If i, i′ : F → ℝ³ are two regularly homotopic generic immersions, then it has been shown in [5] 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. For F orientable we show that for any generic immersion i : F → ℝ³ and any diffeomorphism h : F → F such that i and i º h are regularly homotopic,

Q(i, i º h) = (rank(h_* − Id) + (n + 1)∊(h)) mod 2,

where h_* is the map induced by h on H₁(F, ℤ/2), n is the genus of F and ∊(h) is 0 or 1 according to whether h is orientation preserving or reversing, respectively.

We then give an explicit formula for Q(e, e′) for any two regularly homotopic embeddings e, e′ : F → ℝ³. The formula is in terms of homological data extracted from the two embeddings.