Kahmeyer, Liam2023-04-172023-04-172023https://hdl.handle.net/2097/43076In 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.en-US© the author. This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).http://rightsstatements.org/vocab/InC/1.0/TopologyHomotopyGeometryManifoldsSingularitiesInvariantDissertation