Uniform Morse lemma and isotopy criterion for Morse functions on surfaces / E. A. Kudryavtseva. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2009. № 4. P. 13-22
[Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 150-158].
Let M be a smooth compact (orientable or not) surface with or
without a boundary. Let {\cal D}_0\subset{\rm Diff}(M) be the group of
diffeomorphisms homotopic to {\rm 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{\cal D}_0 and h_2\in{\rm 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 {\rm 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.
