Stellar theory for flag complexes (Q2795650)

The \textit{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.