Uniform Morse lemma and isotopy criterion for Morse functions on surfaces

Let M be a smooth compact (orientable or not) surface with or without a boundary. Let D 0 ⊂ Diff(M) be the group of diffeomorphisms homotopic to id M. Two smooth functions f, g: M → ℝ are called isotopic if f = h 2 g h 1 for some diffeomorphisms h 1 ∈ D 0 and h 2 ∈ 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. © 2009 Allerton Press, Inc.