A homotopy invariant of stable maps to oriented surfaces

Abstract

The singular set of a generic map $f: M\to F$ of a manifold $M$ of dimension $m\ge 2$ to an oriented surface $F$ is a closed smooth curve $Σ(f)$. We study the parity of the number of components of $Σ(f)$. The image $f(Σ)$ of the singular set inherits canonical local orientations via so-called chessboard functions. Such a local orientation gives rise to the cumulative winding number $ω(f)\in \frac{1}{2}\mathbb{Z}$ of $Σ(f)$. When the dimension of the manifold $M$ is even we also define an invariant $I(f)$ which is the residue class modulo $4$ of the sum of the number of components of $Σ(f)$, the number of cusps, and twice the number of self-intersection points of $f(Σ)$. Using the cumulative winding number and the invariant $I(f)$, we show that the parity of the number of connected components of $Σ(f)$ does not change under homotopy of $f$ provided that one of the following conditions is satisfied: (i) the dimension of $M$ is even, (ii) the singular set of the homotopy is an orientable manifold, or (iii) the image of the singular set of the homotopy does not have triple self-intersection points.