2017
Том 69
№ 9

All Issues

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

Feshchenko B. G., Maksimenko S. I.

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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.

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