On an extension problem on the moment curve

TL;DR

This paper investigates the extension of finite simplicial complexes on the moment curve into triangulations of cyclic polytopes in low dimensions, revealing new algebraic applications in higher dimensions.

Contribution

It establishes the existence of such triangulations for dimensions 2 to 4 and constructs counterexamples for dimensions 5 and above, linking geometric and algebraic concepts.

Findings

Triangulations exist for 2-4 dimensions.

Counterexamples for dimensions 5 and higher.

Application to cluster categories and algebraic structures.

Abstract

We show that for $2\le d\le 4$, every finite geometric simplicial complex $\Delta$ in $\mathbb{R}^d$ with vertices on the moment curve can be extended to a triangulation $T$ of the cyclic polytope $C$ where $\Delta, T$ and $C$ all have the same vertex set. Further, for $d\ge 5$ we construct for every $n\ge d+3$ complexes $\Delta$ 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 $A$.