Homotopy properties of spaces of smooth functions on 2-torus

Abstract

Let $f:T^2\to \mathbb{R}$ be a Morse function on a 2-torus, $\mathcal{S}(f)$ and $\mathcal{O}(f)$ be its stabilizer and orbit with respect to the right action of the group $\mathcal{D}(T^2)$ of diffeomorphisms of $T^2$, ${\mathcal{D}}_{\mathrm{id}}(T^2)$ be the identity path component of $\mathcal{D}(T^2)$, and ${\mathcal{S}}^{\prime }(f)=\mathcal{S}(f)\cap {\mathcal{D}}_{\mathrm{id}}(T^2)$. We give sufficient conditions under which $${\pi }_1{\mathcal{O}}_f(f)\cong {\pi }_1\mathcal{D}(T^2)\times {\pi }_0{\mathcal{S}}^{\prime }(f)\equiv {\mathbb{Z}}^2\times {\pi }_0{\mathcal{S}}^{\prime }(f).$$ In fact this result holds for a larger class of smooth functions $f:T^2\to \mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smothly equivalent to a homogeneous polynomial ${\mathbb{R}}^2\to \mathbb{R}$ without multiple factors.

8 pages, 2 figures

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