Shellable complex

The next theorem provides a handy criterion for being able to conclude that an induced subcomplex (see Definition 2.40) of a shellable complex is shellable as well. 

The conditions of Theorem 12.5 are reasonably restrictive. They have to be, since most subcomplexes of the shellable complexes are not shellable. However, it turns that in combinatorial situations these conditions are often satisfied in a natural way. 

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. 

Our presentation centers on the lexshellable posets, as the most general form of lexicographic shellability. We start with the classical situation of order complexes of posets, and then proceed to describe how this generalizes to... 

The next theorem smmnarizes the most important properties of a shellable generalized simplicial complex. 

Theorem 12.3. Assume that A is a shellable generalized simplicial complex, with Fi, F2,..., Ft being the corresponding shelling order of the maximal simplices, and E being the set of spanning simplices. Then the following facts hold ... 

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. 

In this case, the generalized simplicial complex /1[S] is shellable as well, and a shelling order on its maximal simplices is induced by any shelling order on A. 

One standard situation in which shellability has often been used is the study of the order complexes of partially ordered sets. Classically, posets whose order complexes are shellable are themselves called shellable. There is, however, no difEculty whatsoever to extend the framework of shellability to encompass the case of nerves of acyclic categories, which are generalized simplicial complexes as well. 

The following condition is sufficient to guarantee shellability of the order complex. 

Proposition 12.9. Let P be an EL-shellahle poset. Then the simplicial complex A P) is shellable. Moreover, the spanning simplices corresponding to the induced lexicographic shelling order are indexed by the weakly decreasing chains. 

For the sake of simplicity we have so far discussed lexicographic shellability in the context of order complexes of posets. It turns out that working in the generality of nerves of acyclic categories does not cause any substantial problems, and essentially everything can be extended to this context. 

Sha02] J. Shareshian, On the shellability of the order complex of the subgroup lattice of a finite group, Trans. Amer. Math. Soc. 353 (2001), no. 7, pp. 

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