Speaker: Liam Kahmeyer (KSU)

Abstract: 

In 2019, Osamu Saeki showed that for two homotopic generic fold maps f,g:S^3 --> S^2 with respective singular sets \Sigma(f) and \Sigma(g) whose respective images f(\Sigma) and g(\Sigma) are smoothly embedded, the number of components of the singular sets, respectively denoted #|\Sigma(f)| and #|\Sigma(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 > 1 to an oriented surface N of finite genus with f(\Sigma) smoothly embedded, under what conditions (if any) is #|\Sigma(f)| a Z/2-homotopy invariant? The goal of this talk is to explore this question. Namely, I will show that for smooth generic fold maps f:M --> N of a smooth closed oriented manifold M of dimension m > 1 to an oriented surface N of finite genus with f(\Sigma) smoothly embedded, #|\Sigma(f)| is a modulo two homotopy invariant provided one of the following conditions is satisfied: (a) dim(M) = 2q for q > 0, (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.