Discrete Morse Theory

Beyond the encoding of all allowed collapsing orders as the set of linear extensions of the universal object U (P, M), viewing the posets with small fibers as the central notion of the combinatorial part of discrete Morse theory is also invaluable for the structural explanation of a standard way to construct acyclic matchings as unions of acyclic matchings on fibers of a poset map. 

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

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

In this section we give a version of discrete Morse theory that is adapted to the setting of arbitrary free chain complexes. 

Discrete Morse theory is a tool that was discovered by Forman, whose original article [For98], as well as a later survey [For03], are warmly recommended as excellent sources of background information, as well as some topics that we did not cover in this chapter. 

Lexicographic shellability is an important tool for studying the topological properties of the order complexes of partially ordered sets. Although, as we shall see in Remark 12.4, discrete Morse theory is more powerful as a method, shellability may still be useful in concrete applications. We take a detailed look at this concept in this section. 

Remark 12.f. As mentioned above discrete Morse theory is more powerful as a method than shellability. The rationale for this fact is provided by Theorem 12.3(1), saying that the complex A = A IJ j Into- is collapsible, coupled with the fact that a generalized simplicial complex is collapsible if and only if there exists an acyclic matching on the set of its simplices see Theorem 11.13(a) and Remark 11.14. 

Instead of verif3dng that the sequence of coUap>ses is correct in the last paragraph of the proof, we could simply notice that the defined matching is acyclic and derive the result by discrete Morse theory see Theorem 11.13. 

Proof. We give a direct combinatorial matching argument using the chain complex version of the discrete Morse theory from Section 11.3. 

It is now crucial to realize that the same collapses can still be performed in the cochain complex A, d ), even though we are working over the integers. This follows from the version of discrete Morse theory for chain complexes from Section 11.3, since our matching is acyclic and the weights on the matching relations, which here are the incidence numbers, are 1, hence invertible over Z. 

For03] R. Forman, A user s guide to discrete Morse theory, Sem. Lothar. Corn-bin. 48, (2002). 

JW05] M. JoUenbeck, V. Welker, Resolution of the residue class field via algebraic discrete Morse theory, preprint 2005. arXiv math.AC/0501179... 

Ko05c] D.N. Kozlov, Discrete Morse theory for free chain complexes, C. R. Math. 

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