Barycentric subdivision

Definition 2.21. Let A be an abstract simplicial complex. The barycentric subdivision of A is also an abstract simplicial complex, which is denoted by Bd i4 and defined by... 

While taking barycentric subdivision is useful in many situations, sometimes it just produces too many simplices. The next definition provides a more economic, local construction. 

For example, if [Pg.15]

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. 

In particular, Proposition 2.23 suggests one standard way to view the barycentric subdivision as a sequence of stellar ones simply start by taking the stellar subdivisions of the simplices of top dimension, then take the stellar subdivisions of the simplices of dimension one less, and so on, until reaching the vertices. 

One way to think about the barycentric subdivision, which can come in handy in certain situations, is the following. First, we define the barycentric subdivision in the standard n-simplex. By definition, it is the simplicial complex obtained by stratifying the standard n-simplex with the intersections with the hjqjerplanes x = Xj, for 1 [Pg.24]

When the abstract simplicial complex A is finite, then we can take its standard geometric realization and take the barycentric subdivisions of the individual n-simplices as just described. This gives the geometric realization of the barycentric subdivision of A. When, on the other hand, the abstract simplicial complex A is infinite, we can take the barycentric subdivisions of its simplices before the gluing and then observe that the gluing process is compatible with the new cell structure hence we will obtain the geometric realization of the barycentric subdivision of A as well. 

An example of a polyhedral complex is shown on the left of Figure 2.3. Again, all the basic operations and terminology of the simphcial complex extend to the polyhedral ones. This includes the barycentric subdivision, since it can be done on the polyhedra before the gluing, and then one can observe that the obtained complexes will glue to a simphcial complex in a compatible way. 

Since the barycentric subdivision of the generahzed simplicial complex is a geometric realization of an abstract simplicial complex, we can be sure that after taking the barycentric subdivision twice, the trisp will turn into the geometric realization of an abstract simplicial complex. On the other hand, with many trisps, taking the barycentric subdivision once would not suffice for that purpose. 

Proof. To start with, since the barycentric subdivision can be represented as a sequence of stellar subdivisions, see Subsection 2.1.5, it is enough to find a formal deformation leading from X to sd(X, [Pg.95]

If desired, the T-action can be made to be simplicial by passing to the barycentric subdivision (cf. [Bre72, Hat02]). For the interested reader we remark here that sometimes one takes the barycentric subdivision even if the original action already is simplicial. The main point of this is that one can make the action enjoy an additional property if a simplex is preserved by one of the group elements, then it must he pointwise fixed by this element. 

Definition 10.10. For an arbitrary acyclic category C, let BdC denote the poset whose minimal elements are objects ofC, and whose other elements are all composahle morphism chains consisting of nonidentity morphisms of C. The elementary order relations are given by composing morphisms in the chain, and by removing the first or the last morphism. The poset BdC is called the barycentric subdivision of C. 

In Figure 10.7 we show an acyclic category and its barycentric subdivision, whereas in Figure 10.8 we show corresponding nerves. 

In particular, as an immediate corollary of (10.5) we see that an arbitrary regular CW complex is homeomorphic to the order complex of some poset. We also obtain an alternative way to define the barycentric subdivision of... 

Example 10.12. Let P be a totally ordered set with n elements. Clearly, its barycentric subdivision is the Boolean algebra with the minimal element removed BdP = 6. Equation (10.7) implies that... 

Let us now consider the graph associated to the barycentric subdivision of X. In our notation, this graph is called Gbux- We claim that the neighborhood complex of this graph Af GBdx) is homotopy equivalent to BdX, and... 

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