Comptes Rendus
no. 12
Homological Algebra
Discrete Morse theory for free chain complexes

Presented by: Christophe Soulé

Dmitry N. Kozlov1
1 Eidgenössische Technische Hochschule, Zürich, Switzerland
Comptes Rendus. Mathématique, Volume 340 (2005) no. 12, pp. 867-872.

We extend the combinatorial Morse complex construction to arbitrary free chain complexes, and give a short, self-contained, and elementary proof of the quasi-isomorphism between the original chain complex and its Morse complex. Even stronger, the main result states that, if C∗ is a free chain complex, and M an acyclic matching, then C∗=C∗M⊕T∗, where C∗M is the Morse complex generated by the critical elements, and T∗ is an acyclic complex.

On étend la construction du complexe de Morse combinatoire aux complexes de chaînes libres généraux, et on donne une démonstration brève et élémentaire du quasi-isomorphisme entre le complexe de chaînes original et son complexe de Morse. Plus précisément, le résultat principal dit que, si C∗ est un complexe de chaînes libres et M est une correspondance acyclique, alors C∗=C∗M⊕T∗, où C∗M est le complexe de Morse engendré par les éléments critiques et T∗ est un complexe acyclique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.04.036

Dmitry N. Kozlov 1

1 Eidgenössische Technische Hochschule, Zürich, Switzerland 
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Dmitry N. Kozlov. Discrete Morse theory for free chain complexes. Comptes Rendus. Mathématique, Volume 340 (2005) no. 12, pp. 867-872. doi : 10.1016/j.crma.2005.04.036. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.04.036/

[1] R. Forman Morse theory for cell complexes, Adv. Math., Volume 134 (1998) no. 1, pp. 90-145

[2] D.N. Kozlov Collapsibility of Δ(Πn)/Dn and some related CW complexes, Proc. Amer. Math. Soc., Volume 128 (2000) no. 8, pp. 2253-2259

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