The combinatorics of gradient-like flows and foliations on closed surfaces. I: Topological classification (Q1873292)

The author constructs complete topological invariants for gradient-like flows on closed orientable surfaces, and for gradient-like flows on non-orientable surfaces. The invariants in the orientable case are given in terms of ``special rotation graphs'', and in the non-orientable case in terms of ``embedding schemes'' associated with the flows. The special rotation graph of a flow is a graph whose vertices are the sources and the saddle points and whose edges are the separatrices joining the sources with the saddle points. In the non-orientable case there are extra data on the graph which are defined as a marking on the edges. The author also obtains a topological invariant for Morse-Smale foliations without closed orbits (called gradient-like foliations) on closed oriented smooth connected surfaces, which he calls the ``current graph'' of the foliation. The author makes a relation of these invariants with the classical invariants of Peixoto for Morse-Smale flows on surfaces.