A homotopy invariant of image simple fold maps to oriented surfaces

In 2019, Osamu Saeki showed that for two homotopic generic fold maps $f,g:S^3 \rightarrow S^2$ 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 $f:M \rightarrow N$ of a smooth manifold $M$ of dimension $m \geq 2$ to an oriented surface $N$ of finite genus with $f(\Sigma)$ smoothly embedded, under what conditions (if any) is $\#|\Sigma(f)|$ a $\mathbb Z/2$-homotopy invariant? The goal of this talk is to explore this question. Namely, I will show that for smooth generic fold maps $f:M \rightarrow N$ of a smooth closed oriented manifold $M$ of dimension $m\geq 2$ 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) $\textrm{dim}(M) = 2q$ for $ q \geq 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.

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