Elena A. Kudryavtseva

The topology of spaces of morse functions on surfaces

The topology of the space F = F(M) of Morse functions on a compact smooth orientable two-dimensional surface M is studied.

Uniform Morse lemma and isotopy criterion for Morse functions on surfaces

AbstractLet M be a smooth compact (orientable or not) surface with or without a boundary. Let

Special framed Morse functions on surfaces

\mathcal{D}_0

Indecomposable branched coverings over the projective plane by surfaces M with χ(M) ≤ 0

⊂ Diff(M) be the group of diffeomorphisms homotopic to idM. Two smooth functions f, g: M → ℝ are called isotopic if f = h2 ℴ g ℴ h1 for some diffeomorphisms h1 ∈

Topology of the spaces of functions with prescribed singularities on surfaces

\mathcal{D}_0

Maximally symmetric cell decompositions of surfaces and their coverings

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 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.