Reconstructing Polytopes and Pseudomanifolds

TL;DR

This paper establishes new results on the determination of polytopes and pseudomanifolds by their face incidences, solving open problems and extending understanding in combinatorial and topological structures.

Contribution

It proves that 4-polytopes are determined by edge incidences, shows limitations of skeleton-based determination in higher dimensions, and extends results to homology manifolds and pseudomanifolds.

Findings

4-polytopes are determined by edge-polygon incidences

Not all d-polytopes are determined by their (d-3)-skeletons

Certain homology manifolds are determined by specific face incidences

Abstract

We prove that every 4-polytope is determined by its edge-polygon incidences, solving an open problem of Gr\"unbaum. For each $d \geq 3$, we show that not every $d$-polytope is determined by its $(d-3)$-skeleton and dual $(d-3)$-skeleton together, answering a question of Samper. In the simplicial realm, we prove that for $d \geq 4$ and $\lceil \frac{d}{2} \rceil \leq k \leq d-2$, every homology $(d-1)$-manifold is determined by the incidences of its $k$- and $(k-1)$-faces. For $d \geq 5$ and $\lceil \frac{d+1}{2} \rceil \leq k \leq d-2$, we extend our proof to normal $(d-1)$-pseudomanifolds whose $(2d-2k-1)$-dimensional links are homology manifolds. Finally, we prove that not every normal $(d-1)$-pseudomanifold is determined by its $(d-2)$-skeleton.