A uniform Morse lemma and a criterion of isotopy of Morse functions on surfaces
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 о g о h1 for some diffeomorphisms h1 ∈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 neighbourhoods 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.