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In the case of smooth manifolds, we use Forman's discrete Morse theory torealize combinatorially any Thom-Smale complex coming from a smooth Morsefunction by a couple triangulation-discrete Morse function. As an application,we prove that any Euler structure on a smooth oriented closed 3-manifold has aparticular realization by a complete matching on the Hasse diagram of atriangulation of the manifold.

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