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}
}
Gallais, Étienne. 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/
[1] , Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007), 243–255. | MR 2320156 | Zbl 1234.57013
[2] , On discrete Morse functions and combinatorial decompositions, Formal power series and algebraic combinatorics (Vienna, 1997), Discrete Math. 217 (2000), 101–113. | MR 1766262 | Zbl 1008.52011
[3] and , Discrete Morse theory and graph braid groups, Algebr. Geom. Topol. 5 (2005), 1075–1109. | MR 2171804 | Zbl 1134.20050
[4] , Combinatorial vector fields and dynamical systems, Math. Z. 228 (1998), 629–681. | MR 1644432 | Zbl 0922.58063
[5] , Morse theory for cell complexes, Adv. Math. 134 (1998), 90–145. | MR 1612391 | Zbl 0896.57023
[6] , Witten-Morse theory for cell complexes, Topology 37 (1998), 945–979. | MR 1650414 | Zbl 0929.57020
[7] and , “-Manifolds and Kirby Calculus”, Graduate Studies in Mathematics, Vol. 20, American Mathematical Society, Providence, RI, 1999. | MR 1707327 | Zbl 0933.57020
[8] and , Computing optimal Morse matchings, SIAM J. Discrete Math. 20 (2006), 11–25. | MR 2257241 | Zbl 1190.90162
[9] , and , Optimal discrete Morse functions for 2-manifolds, Comput. Geom. 26 (2003), 221–233. | MR 2005300 | Zbl 1031.65031
[10] , and , Applications of Forman’s discrete Morse theory to topology visualization and mesh compression, Transactions on Visualization and Computer Graphics 10 (2004), 499–508.
[11] and , “The Topology of Complexes”, Van Nostrand Reinhold Co., New York, 1969. | Zbl 0207.21704
[12] , “Morse Theory”, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. | MR 163331
[13] , “Lectures on the -Cobordism Theorem”, Notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, N.J., 1965. | MR 190942
[14] and , Holomorphic disks and knot invariants, Adv. Math 1 (2004), 58–116. | MR 2065507 | Zbl 1062.57019
[15] and , Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), 1027–1158. | MR 2113019 | Zbl 1073.57009
[16] , On the Novikov complex for rational Morse forms, Ann. Fac. Sci. Toulouse Math. (6) 4 (1995), 297–338. | Numdam | MR 1344724
[17] and , “Introduction to Piecewise-Linear Topology”, Springer-Verlag, New York, 1972. | MR 350744 | Zbl 0254.57010
[18] and , Combinatorial Morse theory and minimality of hyperplane arrangements, Geom. Topol. 11 (2007), 1733–1766. | MR 2350466 | Zbl 1134.32010
[19] , Morse theory from an algebraic viewpoint, Trans. Amer. Math. Soc. 358 (2006), 115–129 (electronic). | MR 2171225 | Zbl 1150.16008
[20] , Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 607–643, 672. | MR 1013714
[21] , On -complexes, Ann. of Math. (2) 41 (1940), 809–824. | JFM 66.0955.03 | MR 2545
[22] , “Geometric Integration Theory”, Princeton University Press, Princeton, New Jersey, 1957. | MR 87148 | Zbl 0083.28204
