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 $\mathcal{S}'(f) = \mathcal{S}(f) \cap \mathcal{D}_{\mathrm{id}}(T^2)$. We give sufficient conditions under which $$ \pi_1\mathcal{O}_f(f) \ \cong \ \pi_1\mathcal{D}(T^2) \times \pi_0 \mathcal{S}'(f) \ \equiv \ \mathbb{Z}^2 \times \pi_0 \mathcal{S}'(f).$$ 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.