Discrete Morse theory for free chain complexes

We extend the combinatorial Morse complex construction to arbitrary free chain complexes, and give a short, self-contained, and elementary proof of the quasi-isornorphism 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) Theta T-* where C-*(M) is the Morse complex generated by the critical elements, and T* is an acyclic complex. (c) 2005 Academie des sciences. Published by Elsevier SAS. All rights reserved.