Discrete Morse Theory for CW Complexes

The role of the base space here is played by the poset Q, and the fiber maps gq are given by the acyclic matchings on the subposets ip q). The decomposition theorem tells us that there exists a poset map from P to the total space of the corresponding poset fibration, and that the fibers of this map are the same as the fibers of the fiber maps Qq. Since the latter are given by acyclic matchings, we conclude that we have a poset map from P with small fibers that corresponds precisely to the patching of acyclic matchings on the subposets 95 (g), for q Q.  

We conclude our discussion of poset maps with small fibers by mentioning that this point of view yields a rich class of generalizations. Indeed, any choice of the set of allowed fibers will yield a combinatorial theory that could be interesting to study. One could, for instance, allow any Boolean algebra as a fiber. This would correspond to the theory of all coUapses, not just the elementary ones, which we get when considering the small fibers. One can take any other infinite family of posets. One prominent family is that of partition lattices IIn =i- What happens if we consider all poset maps with partition lattices as fibers  

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The Main Theorem of Discrete Morse Theory for CW Complexes... 

Main theorem of discrete Morse theory for CW complexes)... 

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