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 $\mathcal{D}_0\subset\operatorname{Diff}(M)$ be the group of diffeomorphisms homotopic to $\operatorname{id}_M$. Two smooth functions $f,g : M\to\mathbb{R}$ are called isotopic if $f=h_2\circ g\circ h_1$ for some diffeomorphisms $h_1\in\mathcal{D}_0$ and $h_2\in\operatorname{Diff}^+(\mathbb{R})$. 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 $\operatorname{Diff}(M)$-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.