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Stellar theory for flag complexes. (English) Zbl 1350.57027
The stellar subdivision at the face $$F$$ of an abstract simplicial complex $$\Delta$$ is obtained by replacing the star of $$F$$ with the join of a point not in $$\Delta$$, the boundary of $$F$$, and the link of $$F$$. The authors give an algorithm for sequentially computing the barycentric subdivision of a simplicial complex, where each step only involves the stellar subdivision of an edge. This algorithm leads to an alternate proof of a result of Alexander: if $$\Delta$$ and $$\Gamma$$ are two piecewise linearly homeomorphic simplicial complexes, then there is a sequence of simplicial complexes from $$\Delta$$ to $$\Gamma$$, such that any two adjacent terms in the sequence are related by a single stellar edge subdivision. The authors of the article show that if $$\Delta$$ and $$\Gamma$$ are both flag complexes, then each term in the sequence can also be taken to be a flag complex. It is then shown how this algorithm can be used to randomly search the space of all piecewise linearly homeomorphic simplicial flag spheres of a given dimension.

##### MSC:
 57Q05 General topology of complexes
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