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},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {2},
year = {2010},
pages = {229-252},
zbl = {1201.57026},
mrnumber = {2731156},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_2_229_0}
}
Combinatorial realization of the Thom-Smale complex via discrete Morse theory. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 2, pp. 229-252. http://www.numdam.org/item/ASNSP_2010_5_9_2_229_0/
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