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 Ff 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) → Zp+r-1, in particular |π0(F)| = ∞ and the component Ff 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 idM in D, and Df⊂ D the set of diffeomorphisms preserving Ff. Let Hf be the subgroup of Df generated by D0 and all diffeomorphisms h∈ D preserving some functions f1∈ Ff, and let Hfabs be its subgroup generated by D0 and the Dehn twists about the components of level curves of functions f1∈Ff. We prove that Hfabs⊊Df if q ≥ 2, and construct an epimorphism Df / Hfabs → Z2q-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) → Df / Hf and finite generating sets for the groups Df / D0 and ;Df / Hf 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.