Structure of the fundamental groups of orbits of smooth functions on surfaces

TL;DR

This paper investigates the fundamental group structure of the orbit space of Morse functions on certain orientable surfaces, extending previous homotopy analyses to include the fundamental group for a broader class of smooth maps.

Contribution

It describes the structure of the fundamental group of the orbit of Morse functions on orientable surfaces, generalizing earlier results to maps with specific local polynomial properties.

Findings

Determined the structure of     for certain surfaces.

Extended analysis to a broader class of smooth maps.

Provided new insights into the topology of orbit spaces of Morse functions.

Abstract

Let $M$ be a smooth compact connected surface, $P$ be either the real line $\mathbb{R}$ or the circle $S^1$ and $f:M\to P$ be a Morse map. Denote by $\mathcal{S}(f)$ and $\mathcal{O}(f)$ the corresponding stabilizer and orbit of $f$ with respect to the right action of the group $\mathcal{D}(M)$ of diffeomorphisms of $M$. In a series of papers the author described homotopy types of $\mathcal{S}(f)$ and computed higher homotopy groups of $\mathcal{O}(f)$. The present paper describes the structure of the remained fundamental group $\pi_1 \mathcal{O}(f)$ for the case when $M$ is orientable and differs from $2$-sphere and $2$-torus. The result holds as well for a larger class of smooth maps $f:M\to P$ having the following property: the germ of $f$ at each of its critical points is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to\mathbb{R}$ without multiple factors.