Geometric bistellar moves relate geometric triangulations (Q2216669)

A triangulation \(K\) of a Riemannian manifold \(M\) is called a \textit{geometric triangulation} if the interior of each simplex (of \(K\)) is totally geodesic. Here all the triangulations are simplicial, i.e., each simplex \(\alpha\) has \(\dim(\alpha)+1\) vertices and intersection of any two simplices is a face of both. Suppose \(K_1\) and \(K_2\) are two geometric triangulations of a Riemannian manifold \(M\). If \(K_2\) is obtained from \(K_1\) by inserting a new vertex \(v\) in the interior of a simplex \(\alpha\) of \(K_1\), i.e., \(K_2 = (K_1\setminus (\alpha\ast\mathrm{lk}_{K_1}(\alpha))\cup (v\ast \partial\alpha\ast\mathrm{lk}_{K_1}(\alpha))\) then \(K_2\) is said to be obtained from \(K_1\) by a \textit{stellar subdivision} on \(\alpha\). Two geometric triangulations \(K_1\), \(K_2\) of a Riemannian \(d\)-manifold \(M\) are said to differ in a \textit{bistellar flip} if there are simplices \(\sigma_1\in K_1\) and \(\sigma_2\in K_2\) such that the stellar subdivisions of \(K_1\) on \(\sigma_1\) and \(K_2\) on \(\sigma_2\) produce the same simplicial complex. Observe that, in that case, \(K_2 = (K_1\setminus (\sigma_1\ast \partial \sigma_2))\cup (\sigma_2\ast\partial \sigma_1)\). If \(\dim(\sigma_1) =i\) then the bistellar operation from \(K_1\) to \(K_2\) is said to be a \textit{bistellar flip of type \((i+1, d-i+1)\)} (or a \textit{bistellar \((d-i)\)-move}). So, a bistellar 0-move is a stellar operation on a \(d\)-simplex. For a geometric triangulation \(K\) of a Riemannian manifold \(M\), a \textit{derived subdivision} \(\beta K\) of \(K\) is obtained from \(K\) by stellarly subdividing \(K\) at all simplices of \(K\) in order of decreasing dimension. In this article, the authors prove the following: (i) If \(K_1\), \(K_2\) are two geometric triangulations of a compact Riemannian manifold \(M\) of constant curvature and \(L\) is a common subcomplex of \(K_1\), \(K_2\) with \(|L|\supset \partial M\), then for some non-negative integer \(s\), the \(s\)-th derived subdivisions \(\beta^sK_1\) and \(\beta^sK_2\) are related by geometric bistellar moves which keep \(\beta^sL\) fixed. (ii) If two geometric triangulations \(K_1\), \(K_2\) of a cusped hyperbolic manifold have a common geometric subdivision then for some non-negative integer \(s\), the \(s\)-th derived subdivisions \(\beta^sK_1\) and \(\beta^sK_2\) are related by geometric bistellar moves. (iii) Let \(K_1\), \(K_2\) be two geometric triangulations of a closed 3-manifold \(M\) of constant curvature. Moreover, the diameter of the star of each simplex is less than \(\pi\) if \(M\) is spherical. Then \(K_1\) and \(K_2\) are related by geometric bistellar moves.