The Main Theorem of Algebraic Morse Theory

The main theorem of algebraic Morse theory brings to light a certain structure in (T. Namely, by choosing a different basis, one can represent C as a direct sum of two chain complexes, of which one is a direct sum of atom chain complexes, in particular acyclic, and the other one is isomorphic to C. For convenience, the choice of basis can be performed in several steps, one step for each matched pair of the basis elements. 

Assume that we have a free chain complex with a basis (C, Q), and an acyclic matching Ai. Then C decomposes as a direct sum of chain complexes 0 %, where T Atom (dim6). 

It can be advisable to use the example in Subsection 11.3.3 as an illustration for the following proof. 

To start with, let us choose a linear extension L of the partially ordered set P C, f2) satisfying the conditions of Theorem 11.2, and let l denote the corresponding total order. 

Assume first that C is boimded without loss of generality, we can assume that Ci = 0 for i C 0, and t AT. Let rn — Af denote the size of the matching, and let / = 1 — 2m denote the munber of critical cells. 

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