Free chain complex with a basis

In order to introduce a combinatorial element into this setting, we need to choose a basis (i.e., a set of free generators) for each C . When this is done, we say that we have chosen a basis i = (J S7 for the entire chain complex C. We write (C, 1 ) to denote a chain complex with a basis. A free chain complex with a basis is the main object of study of algebraic Morse theory. 

Definition 11.23. Let (C, fi) be a free chain complex with a basis, and let A4 be an acyclic matching. The Morse complex... 

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). 

Let X be an arbitrary CW complex. It is easy to see that for every A > 0, the relative homology group Hk X X X) is free abelian, with a basis indexed by the fe-cells of X. Hence, we can use the group as an alternative definition for the cellular chain group The cellular... 

The absence of mesomorphism in these compounds was explained on the basis of space-filling requirements. Thus, the intercalation of pyrazine between the binuclear units creates free volume which needs to be filled to obtain a stable, condensed phase when the carboxylates bear only one chain, the interdimeric space is likely filled by the aliphatic chains belonging to a different polymeric chain, giving rise to a crossed structure which prevents the formation of a columnar mesophase. However, as will be seen later, liquid-crystalline behavior was induced in the case of mixed-valence diruthenium(II,III) carboxylate complexes with bulky equatorial Kgands bearing two and three aliphatic chains as with such ligands, it was possible to fill the interdimeric space and thus to induce a thermotropic columnar mesophase. Very recently, the synthesis, characterization, and mesomorphic properties of pyrazine-polymerized divalent rhodium benzoates have also been reported (99). " Most of these compounds exhibit columnar (Colh, Coir, CoIn) and cubic mesophases with melting transition temperatures close to, or even below, room temperature. 

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