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 is a free chain complex, and an acyclic matching, then , where is the Morse complex generated by the critical elements, and 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 est un complexe de chaînes libres et est une correspondance acyclique, alors , où est le complexe de Morse engendré par les éléments critiques et est un complexe acyclique.
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@article{CRMATH_2005__340_12_867_0, author = {Kozlov, Dmitry N.}, title = {Discrete {Morse} theory for free chain complexes}, journal = {Comptes Rendus. Math\'ematique}, pages = {867--872}, publisher = {Elsevier}, volume = {340}, number = {12}, year = {2005}, doi = {10.1016/j.crma.2005.04.036}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2005.04.036/} }
TY - JOUR AU - Kozlov, Dmitry N. TI - Discrete Morse theory for free chain complexes JO - Comptes Rendus. Mathématique PY - 2005 SP - 867 EP - 872 VL - 340 IS - 12 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2005.04.036/ DO - 10.1016/j.crma.2005.04.036 LA - en ID - CRMATH_2005__340_12_867_0 ER -
Kozlov, Dmitry N. 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. http://archive.numdam.org/articles/10.1016/j.crma.2005.04.036/
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