A homotopy invariant of image simple fold maps to oriented surfaces

Abstract

In 2019, Osamu Saeki showed that for two homotopic generic fold maps f,g : S³ --> S² with respective singular sets Σ(f) and Σ(g) whose respective images f(Σ) and g(Σ) are smoothly embedded, the number of components of the singular sets, respectively denoted #|Σ(f)| and #|Σ(g)|, need not have the same parity. From Saeki’s result, a natural question arises: For generic fold maps f : M --> N of a smooth manifold M of dimension m ≥ 2 to an oriented surface N of finite genus with f(Σ) smoothly embedded, under what conditions (if any) is #|Σ(f)| a Z/2-homotopy invariant? The goal of this dissertation is to explore this question. Namely, we show that for smooth generic fold maps f : M --> N of a smooth closed oriented manifold M of dimension m ≥ 2 to an oriented surface N of finite genus with f(Σ) smoothly embedded, #|Σ(f)| is a modulo two homotopy invariant provided one of the following conditions is satisfied: (a) dim(M) = 2q for q ≥ 1, (b) the singular set of the homotopy is an orientable manifold, or (c) the image of the singular set of the homotopy does not have triple self-intersection points. Finally, we conclude with a few low-dimensional applications of the main results.