Stellar subdivision

For example, if barycentric subdivision can be accomplished by a sequence of stellar subdivisions. 

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. 

After that, the new simplices are added. By the definition of the stellar subdivision together with our induction assumption, these are indexed by the pairs. .., a, cr ),a k) such that the abstract simplicial complex... 

All in all, we see that this description of the simplices that are added or removed when taking the stellar subdivision of (ofc) in Ak-i, together with the induction assumption that the conditions of our claim describe the abstract simplicial complex Ak-i, implies that the conditions of our claim describe the abstract simplicial complex Ak as well, providing the induction step. This finishes the proof of our claim. ... 

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. 

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]

Finally, we see that performing Steps 1 and 2, in this order, will yield a stellar subdivision of X at a, and therefore our description is completed. ... 

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