2022/08/15 by Kahmeyer, Liam, Sadykov, Rustam · 1 citation
#57R45 #58K30 #58K65 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2208.07297
The singular set of a generic map f: M→ F of a manifold M of dimension m≥ 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)∈ (1)/(2)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.