Uniform Morse Lemma and Isotopy Criterion for Morse Functions on Surfaces / E.A. Kudryavtseva // Vestn. Mosk. Univ., Matem. Mekhan. 2009. № 4. P. 13-22 [Moscow Univ. Math. Bulletin. Vol. 64, No 4, 2009. P. 150-158.]
Let M be a smooth compact (orientable or not) surface with or without a boundary. Let D0⊂ Diff(M) be the group of diffeomorphisms homotopic to idM. Two smooth functions f,g: M→R are called isotopic if f=h2˚g˚h1 for some diffeomorphismsh1 ∈ D0 and h2 ∈ Diff+(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∈F, a collection of Morse local coordinates in disjoint circular neighborhoods of its critical points is constructed, which continuously and Diff(M)-equivariantly depends on f in C∞-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.