Uniform Morse lemma and isotopy criterion for Morse functions on surfaces / E. A. Kudryavtseva. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2009. ? 4. P. 13-22 [Moscow Univ. Math. Bulletin. Vol. 64, N 4, 2009. P. 150-158].

Let M be a smooth compact (orientable or not) surface with or without a boundary. Let {\cal D}_0\subset{\rm Diff}(M) be the group of diffeomorphisms homotopic to {\rm id}_M. Two smooth functions f,g : M\to\mathbb R are called isotopic if f=h_2\circ g\circ h_1 for some diffeomorphisms h_1\in{\cal D}_0 and h_2\in{\rm Diff}^+(\mathbb R). Let F be the space of Morse functions on M which are constant on each boundary component and have no critical points on the boundary. A criterion for two Morse functions from F to be isotopic is proved. For each Morse function f\in F, a collection of Morse local coordinates in disjoint circular neighbourhoods of its critical points is constructed, which continuously and {\rm Diff}(M)-equivariantly depends on f in C^\infty-topology on F ("uniform Morse lemma"). Applications of these results to the problem of describing the homotopy type of the space F are formulated.

Key words: Morse function, equivalence of Morse functions, closed surface, Morse lemma.