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.