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

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.