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Discrete Morse Theory for free chain complexes

Dmitry N. Kozlov
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Abstract

We extend the combinatorial Morse complex construction to the 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 $\cm$ an acyclic matching, then $C_*=C_*^\cm\oplus T_*$, where $C_*^\cm$ is the Morse complex generated by the critical elements, and $T_*$ is an acyclic complex.

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Dmitry N. Kozlov. 2005-04-21. Discrete Morse Theory for free chain complexes. https://arxiv.org/abs/cs/0504090

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