Combinatorial realization of the Thom-Smale complex via discrete Morse theory

Gallais, Étienne

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 229-252.

In the case of smooth manifolds, we use Forman’s discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a pair triangulation-discrete Morse function. As an application, we prove that any class of homologous vector fields on a smooth oriented closed 3-manifold can be realized by a perfect matching on the Hasse diagram of a triangulation of the manifold.

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