Graphs and flows on surfaces

In 1971, M. M. Peixoto [15] introduced an important topological invariant of Morse-Smale flows on surfaces, which he called a distinguished graph X* associated with a given flow. Here we show how the Peixoto invariant can be essentially simplified and revised by adopting a purely topological point of view connected with the embeddings of arbitrary graphs into compact surfaces. The newly obtained invariant, X-R, is a rotation of a graph X generated by a Morse-Smale flow. (a rotation R is a cyclic order of edges given in every vertex of X.) The invariant X-R 'reads-off' the topological information carried by a flow, being in a one-to-one correspondence with the topological equivalence classes of Morse-Smale flows double dagger. As a counterpart to the equivalence result we prove a realization theorem for an 'abstractly given' X-R. (Our methods are completely different from those of Peixoto and they clarify the connections between graphs and flows on surfaces.) The idea of 'rotation systems' on graphs can be further exploited in the study of recurrent flows (and foliations) with several disjoint quasiminimal sets on surfaces [10].