On an extension problem on the moment curve

We show that for $2\le d\le 4$, every finite geometric simplicial complex $\mathrm{\Delta }$ in ${\mathbb{R}}^d$ with vertices on the moment curve can be extended to a triangulation $T$ of the cyclic polytope $C$ where $\mathrm{\Delta },T$ and $C$ all have the same vertex set. Further, for $d\ge 5$ we construct for every $n\ge d+3$ complexes $\mathrm{\Delta }$ on $n$ vertices for which no such triangulations $T$ exist. Our result for $d=4$ has the following novel algebraic application, due to a correspondence by Oppermann and Thomas (JEMS, 2012): every maximal rigid object in ${\mathcal{O}}_{A_n^2}$ is cluster tilting, where ${\mathcal{O}}_{A_n^{\delta }}$ denotes a higher dimensional cluster category introduced by Oppermann and Thomas for $A_n^{\delta }$, where $A_n^{\delta }$ denotes a higher Auslander algebra of linearly oriented type $A$.