Discrete Morse Theory for free chain complexes
<|reference_start|>Discrete Morse Theory for free chain complexes: 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.<|reference_end|>
@article{kozlov2005discrete,
    title={Discrete Morse Theory for free chain complexes},
    author={Dmitry N. Kozlov},
    journal={Comptes Rendus Mathematique, Acad. Sci. Paris, Ser. I 340 (2005),
  pp. 867-872},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0504090},
    primaryClass={cs.DM math.RA}
}