Let be a connected compact oriented surface and be either ℝ or . Let () be the space of Morse mappings → with compact-open topology. The classification of path-components of () was independently obtained by S. V. Matveev and V. V. Sharko for the case = ℝ, and by the author for = . In this...

We classify the path-components of the space of circle-valued Morse functions on compact surfaces: two Morse functions f,g : → belong to same path-component of this space if and only if they are homotopic and have equal numbers of critical points at each index

Let be a smooth compact surface, orientable or not, with boundary or without it, either the real line R or the circle , and () the group of diffeomorphisms of acting on C∞(M, P) by the rule , for ∈ () and ∈ (, ).Let : → be a Morse function and () be the orbit of f under this action. We...

Let Φ be a flow on a smooth, compact, finite-dimensional manifold . Consider the subset D(Φ) of (, ) consisting of diffeomorphisms of preserving the foliation of the flow Φ. Let also D(Φ) be the identity path component of D(Φ) with compact-open topology. We prove that under mild conditions...