Generalized simplicial complex

Definition 2.41. A polyhedral complex whose cells are simplices is called a generalized simplicial complex. ... 

This is a class of polyhedral complexes that contains the generalized simplicial complexes and is closed under direct products. An example of a prod-simphcial complex is shown on the right of Figure 2.3. 

Definition 6.13. Let A be a generalized simplicial complex. Leta,r S A such that... 

When AI and A2 are two generalized simplicial complexes such that there exists a sequence of collapses leading from to A2, we shall use the notation... 

Theorem 6.16. A generalized simplicial complex A is contractible if and only if there exists a sequence of collapses and expansions (operation inverse to the collapse, also called an anticollapse) leading from A to a vertex. 

Definition 6.17. Two generalized simplicial complexes are said to have the same simple homotopy type if there exists a sequence of elementary collapses and expansions leading from one to the other. Such a sequence is called a formal deformation. 

Theorem 6.16 is a special case of the fundamental theorem that in particular says that two simply connected generalized simplicial complexes are homotopy equivalent if and only if they have the same simple homotopy type. 

It is known that a subdivision of any generalized simplicial complex X has the same simple homotopy type as X. Let us show a special case of that. 

Discrete Configuration Spaces of Generalized Simplicial Complexes... 

Definition 9.27. For any generalized simplicial complex A and arbitrary graphs T and G, we set... 

When it is more appropriate, e.g., in Chapter 12, we will forget about the orientations of the simplices of A C) and just view A C) as a generalized simplicial complex. 

Recall from Definition 6.13 that for a generalized simplicial complex A, a sim-plicial collapse is simply a removal of interiors of two simplices cr and r such... 

Fig. 11.11. Examples of labeled forests indexing 2-simplices of the generalized simplicial complex A n )/S .

Note that in Definition 12.1 we use the notion of a generalized simplicial complex. In particular, multiple simplices on the same set of vertices are explicitly allowed. Also, we shall call a generalized simplicial complex pure if all of its maximal simplices have the same dimension. 

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 ... 

The generalized simplicial complex obtained by the removal of the interiors... 

The generalized simplicial complex A is homotopy equivalent to a wedge of spheres that are indexed by the spanning simplices and have corresponding dimensions. More precisely, we have... 

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. 

We shall verify the equivalent shelling condition from Proposition 12.2. Let 1 < a < 6 < m, and consider a simplex a in the intersection Da H Db such that dim a < dimD(, — 2. By our construction, we have ia < and of course, a is a simplex of F, n with dim a < dimF — 2. Therefore, since F, Ft is a shelling order for the generalized simplicial complex A, there must exist j < % such that Fj n contains a simplex r, which in turn contains a as a proper subsimplex. As remarked earlier, this simplex t can... 

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. 

If, in addition, this action satisfies Condition (S2), then the quotient complex is a generalized simplicial complex. 

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