Face poset

A standard combinatorial gadget that one associates to an abstract simplicial complex is that of a face poset. To start with, we have the following definition. 

Definition 2.19. Let A he an arbitrary abstract simplicial complex. A face poset of A is the poset J A) whose set of elements consists of all nonempty simplices of A and whose partial order relation is the inclusion relation on the set of simplices. 

For example, for an arbitrary abstract simplicial complex 2l, a standard linear extension of the face poset P(zi) is obtained by setting a >l t whenever dim <7 > dimr, and choosing an arbitrary order within each set of simplices of the same dimension. 

Proposition 2.23. Let A be an arbitrary finite abstract simplicial complex, and let L be an arbitrary linear extension of the face poset J-(A). Then, the barycentric subdivision Bd A is isomorphic to the abstract simplicial complex obtained from A by a sequence of stellar subdivisions, consisting of one stellar subdivision for every nonempty simplex of A, taking the simplices in decreasing order with respect to the given linear extension. 

An alternative way to think of Definition 9.14 is the following. Assume that we have n data points and N landmark points. Every data point induces an order on the landmark points just sort them with respect to their distances to that point. Every such ordering can be visualized as a path in the Hasse diagram of the Boolean lattice starting from the point nearest to the chosen data point, then proceeding to the union of the two closest ones, then on to the three closest ones, and so on. Now, the witness complex W(A, B) is the maximal abstract sirnplicial complex whose face poset is contained in the union of these paths. 

Just like A, this is a functor from the category of regular CW complexes to the category of posets, since a cellular map between regular CW complexes will induce an order-preserving map between their face posets. 

If n = 3fc + 1, then this preimage is a face poset with a cone with apex in n in particular, the pairing acyclic matching with... 

If n = 3k, we see that X = (p cr) is a face poset of the boundary of a k-dimensional cross-polsdope, which is the same as the fc-fold join of with itself. By Theorem 11.10 the matching constructed up to now is acyclic, and it gives us a collapsing sequence leading to X. In particular, this shows that Ind (Lsk) is homotopy equivalent to... 

If n = 3k, then we again have a face poset of the join of k copies of S°. Denote the sets of vertices of these k copies of S° by xi,yx, xk,yk -Consider the pairing u x, where i is the minimal index such that Vi cr. This is a well-defined acyclic matching with critical cells xi of dimension 0, and yi,..., yk of dimension k... 

Recall that in Subsection 9.1.4, more precisely in Definition 9.9, to an arbitrary graph G we have associated an abstract simplicial complex, called the neighborhood complex, which we denoted by J f G). Note that when A C V G) is a simplex of A/"(G), then so is N A). However, mapping A to N A) would not give a simplicial map from N G) to itself. Instead, we need to proceed to the face poset of Af G), which in our notation is called F Af G)). 

Definition 17.29. We call the cubical complex whose face poset is given by Pn,k o,s above the Kneser cubical complex and denote it by KC k-... 

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