2019
Том 71
№ 6

# Homotopic Properties of the Spaces of Smooth Functions on a 2-Torus

Abstract

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 $${\uppi}_1\mathcal{O}(f)={\uppi}_1{\mathcal{D}}_{\mathrm{id}}\left({T}^2\right)\times {\uppi}_0S^{\prime }(f)\equiv {\mathrm{\mathbb{Z}}}^2\times {\uppi}_0S^{\prime }(f).$$ The obtained result is true for a larger class of functions whose critical points are equivalent to homogeneous polynomials without multiple factors.

English version (Springer): Ukrainian Mathematical Journal 66 (2014), no. 9, pp 1346-1353.

Citation Example: Feshchenko B. G., Maksimenko S. I. Homotopic Properties of the Spaces of Smooth Functions on a 2-Torus // Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1205–1212.

Full text