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

  • S. I. Maksimenko
  • B. G. Feshchenko

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.
Published
25.09.2014
How to Cite
MaksimenkoS. I., and FeshchenkoB. G. “Homotopic Properties of the Spaces of Smooth Functions on a 2-Torus”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, no. 9, Sept. 2014, pp. 1205–1212, http://umj.imath.kiev.ua/index.php/umj/article/view/2211.
Section
Research articles