Reconstructing Polytopes and Pseudomanifolds

We prove that every 4-polytope is determined by its edge-polygon incidences, solving an open problem of Grünbaum. For each [Formula: see text], we show that not every [Formula: see text]-polytope is determined by its [Formula: see text]-skeleton and dual [Formula: see text]-skeleton together, answering a question of Samper. In the simplicial realm, we prove that for [Formula: see text] and [Formula: see text], every homology [Formula: see text]-manifold is determined by the incidences of its [Formula: see text]- and [Formula: see text]-faces. For [Formula: see text] and [Formula: see text], we extend our proof to normal [Formula: see text]-pseudomanifolds whose [Formula: see text]-dimensional links are homology manifolds. Finally, we prove that not every normal [Formula: see text]-pseudomanifold is determined by its [Formula: see text]-skeleton.