Homotopic Properties of the Spaces of Smooth Functions on a 2-Torus - Maksimenko, S. I.; Feshchenko, B. G. (Ukrains’kyi Matematychnyi Zhurnal, 2014)
Let f : T 2 → ℝ be a Morse function on a 2-torus, let S(f) and \( \mathcal{O} \) (f) be, respectively, its stabilizer and orbit with respect to the right action of the group \( \mathcal{D} \) (T 2) of diffeomorphisms of T 2, let \( \mathcal{D} \) id(T 2), be the identity path component of the group \( \mathcal{D} \) (T 2), and let S′(f) = S(f) ∩ \( \mathcal{D} \) id(T 2). We present sufficient conditions under which $$ {\uppi}_1\mathcal{O}(f)={\uppi}_1{\mathcal{D}}_{id}\left(T^2\right)× {\uppi}_0S^{′ }(f)\equiv {\mathrm{\mathbb{Z}}}^2× {\uppi}_0S^{′ }(f). $$ The obtained result is true for a larger class of functions whose critical points are equivalent to homogeneous polynomials without multiple factors.