Reconstructing Polytopes and Pseudomanifolds

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.