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.
@article{ASNSP_2010_5_9_2_229_0, author = {Gallais, \'Etienne}, title = {Combinatorial realization of the Thom-Smale complex via discrete Morse theory}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {229--252}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 9}, number = {2}, year = {2010}, zbl = {1201.57026}, mrnumber = {2731156}, language = {en}, url = {archive.numdam.org/item/ASNSP_2010_5_9_2_229_0/} }
Gallais, Étienne. Combinatorial realization of the Thom-Smale complex via discrete Morse theory. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 229-252. http://archive.numdam.org/item/ASNSP_2010_5_9_2_229_0/
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