Let f : <$EM~symbol О~roman bold R> be a Morse function on a connected compact surface M, and S(f) and O(f) be respectively the stabilizer and the orbit of f with respect to the right action of the group of diffeomorphisms D(M). In a series of papers the first author described the homotopy types of connected components of S(f) and O(f) for the cases when M is either a 2-disk or a cylinder or <$Echi (M)~<<~0>. Moreover, in two recent papers the authors considered special classes of smooth functions on 2-torus T<^>2 and shown that the computations of <$Epi sub 1>O(f) for those functions reduces to the cases of 2-disk and cylinder. In the present paper we consider another class of Morse functions f : <$ET sup 2 ~symbol О~roman bold R> whose KR-graphs have exactly one cycle and prove that for every such function there exists a subsurface <$EQ~symbol <172>~T sup 2> diffeomorphic with a cylinder, such that <$Epi sub 1>O(f) is expressed via the fundamental group <$Epi sub 1 O(f| sub Q )> of the restriction of f to Q. This result holds for a larger class of smooth functions f : <$ET sup 2 ~symbol О~roman bold R> having the following property: for every critical point z of f the germ of f at z is smoothly equivalent to a homogeneous polynomial <$E{ roman bold R} sup 2 ~symbol О~roman bold R> without multiple factors.

• Максименко Сергій Іванович (1974–) (фізико-математичні науки)
• Фещенко Богдан Григорович (фізико-математичні науки)

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