Connected components of spaces of Morse functions with fixed critical points / E. A. Kudryavtseva. //Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2012. ? 1. P. 3-12. [Moscow Univ. Math. Bulletin. Vol. 67, N 1, 2012.].

Let M be a smooth closed orientable surface and F = Fp,q,r be the space of Morse functions on M having exactly p critical points of local minima, q ≥ 1 saddle critical points, and r critical points of local maxima, moreover all the points are fixed. Let F_f be the connected component of a function f∈F in F. By means of the winding number introduced by Reinhart (1960), we construct a surjection π₀(F) → Z^p+r-1, in particular |π₀(F)| = ∞ and the component F_f is not preserved under the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point. Let D be the group of orientation preserving diffeomorphisms of M leaving fixed the critical points, D⁰ be the connected component of id_M in D, and D_f⊂ D the set of diffeomorphisms preserving F_f. Let H_f be the subgroup of D_f generated by D⁰ and all diffeomorphisms h∈ D preserving some functions f₁∈ F_f, and let H_f^abs be its subgroup generated by D⁰ and the Dehn twists about the components of level curves of functions f₁∈F_f. We prove that H_f^abs⊊D_f if q ≥ 2, and construct an epimorphism D_f / H_f^abs → Z₂^q-1, by means of the winding number. A finite polyhedral complex K = Kp,q,r associated to the space F is defined. An epimorphism μ : π₁(K) → D_f / H_f and finite generating sets for the groups D_f / D⁰ and ;D_f / H_f in terms of the 2-skeleton of the complex K are constructed.

Key words: Morse functions on a surface, equivalent and isotopic functions, winding number, Dehn twist, admissible diffeomorphism, polyhedral complex.