A homotopy invariant of stable maps to oriented surfaces

TL;DR

This paper introduces a homotopy invariant for stable maps from manifolds to oriented surfaces, based on the parity of singular set components and winding numbers, which remains unchanged under certain conditions.

Contribution

It defines new invariants, including the cumulative winding number and a residue class invariant, to study the homotopy classes of stable maps to surfaces.

Findings

The parity of the number of singular set components is homotopy invariant under specific conditions.

The cumulative winding number is a key invariant that remains stable during homotopies.

The invariant I(f) captures topological features related to cusps and self-intersections for even-dimensional manifolds.

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 $\Sigma(f)$. We study the parity of the number of components of $\Sigma(f)$. The image $f(\Sigma)$ of the singular set inherits canonical local orientations via so-called chessboard functions. Such a local orientation gives rise to the cumulative winding number $\omega(f)\in \frac{1}{2}\mathbb{Z}$ of $\Sigma(f)$. 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 $\Sigma(f)$, the number of cusps, and twice the number of self-intersection points of $f(\Sigma)$. Using the cumulative winding number and the invariant $I(f)$, we show that the parity of the number of connected components of $\Sigma(f)$ 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.