A uniform Morse lemma and a criterion of isotopy of Morse functions on surfaces
Let <i>M </i>be a smooth compact (orientable or not) surface with or without a boundary. Let <i>D</i><sub><i></i>0</sub> ⊂<i> </i>Diff<i>(M) </i>be the group of diffeomorphisms homotopic to id<sub><i>M</i></sub><i></i>. Two smooth functions <i>f,g : M </i>→ ℝ<i> </i>are called isotopic if <i>f = h</i><sub><i></i>2</sub><i> </i>о<i> g </i>о<i> h</i><sub><i></i>1</sub><i> </i>for some diffeomorphisms h<sub>1</sub> ∈<i>D</i><sub><i></i>0 </sub><i> </i>and h<sub>2</sub> ∈<i> </i>Diff<sup>+</sup>(ℝ). Let <i>F </i>be the space of Morse functions on <i>M </i>which are constant on each boundary component and have no critical points on the boundary. A criterion for two Morse functions from <i>F </i>to be isotopic is proved. For each Morse function <i>f </i>∈<i> F, </i>a collection of Morse local coordinates in disjoint circular neighbourhoods of its critical points is constructed, which continuously and Diff<i>(M</i>)-equivariantly depends on <i>f </i>in <i>C</i><sup><i></i>∞</sup>-topology on <i>F </i>("uniform Morse lemma"). Applications of these results to the problem of describing the homotopy type of the space <i>F </i>are formulated.