Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reconstructing Polytopes and Pseudomanifolds

Joshua Hinman

TL;DR

The paper resolves Grünbaum’s edge–ridge reconstruction question by proving that every $4$-polytope is determined by edge–ridge incidences, and shows that for general $d\ge 3$ not all $d$-polytopes are determined by the $(d-3)$-skeleton together with its dual $(d-3)$-skeleton. It then extends reconstruction phenomena to the simplicial setting: for $d\ge 4$ and $\lceil d/2\rceil \le k \le d-2$, any homology $(d-1)$-manifold is determined by the incidences of its $k$- and $(k-1)$-faces, and this extends to certain normal pseudomanifolds whose $(2d-2k-1)$-dimensional links are homology manifolds. However, not every normal $(d-1)$-pseudomanifold is determined by its $(d-2)$-skeleton, establishing both positive reconstruction results under specific conditions and negative counterexamples in general. The work blends combinatorial polytope theory with topological tools like Alexander duality and Dancis-type reconstruction, providing a unified view of when partial skeleta suffice and when they do not for both polytopes and simplicial complexes.

Abstract

We prove that every 4-polytope is determined by its edge-polygon incidences, solving an open problem of Grünbaum. 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.

Reconstructing Polytopes and Pseudomanifolds

TL;DR

The paper resolves Grünbaum’s edge–ridge reconstruction question by proving that every -polytope is determined by edge–ridge incidences, and shows that for general not all -polytopes are determined by the -skeleton together with its dual -skeleton. It then extends reconstruction phenomena to the simplicial setting: for and , any homology -manifold is determined by the incidences of its - and -faces, and this extends to certain normal pseudomanifolds whose -dimensional links are homology manifolds. However, not every normal -pseudomanifold is determined by its -skeleton, establishing both positive reconstruction results under specific conditions and negative counterexamples in general. The work blends combinatorial polytope theory with topological tools like Alexander duality and Dancis-type reconstruction, providing a unified view of when partial skeleta suffice and when they do not for both polytopes and simplicial complexes.

Abstract

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

Paper Structure

This paper contains 10 sections, 14 theorems, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polytopes and skeleta
  4. CW complexes
  5. Simplicial complexes
  6. Reconstructing a 4-polytope from edges and ridges
  7. The (d-3)-skeleton and dual (d-3)-skeleton are not enough
  8. The (1,d-2)-skeleton is necessary and sufficient
  9. Simplicial complexes
  10. Acknowledgments

Key Result

Theorem 2.5

Let $P,Q$ be $d$-polytopes and $0 \leq a < b < d$. Then any poset isomorphism $\varphi:\mathscr{F}^a(P) \cup \mathscr{F}^b(P) \to \mathscr{F}^a(Q) \cup \mathscr{F}^b(Q)$ extends to an isomorphism $\mathscr{F}_a^b(P) \to \mathscr{F}_a^b(Q)$.

Figures (3)

  • Figure 1: From left to right, polytopes $W$, $P$, and $Q$ for $d=3$. The square base is $F$; the bold edge is introduced when we construct $P$ or $Q$ from $W$.
  • Figure 2: Schlegel diagrams of $P$ (left) and $Q$ (right) for $d=4$. The outer octahedron is $F$; the shaded square is the ridge introduced when we construct $P$ or $Q$ from $W$ (not pictured).
  • Figure 3: Normal $(d-1)$-pseudomanifolds $X$ and $Y$ for $d=3$. In this case, $X$ and $Y$ are manifolds homeomorphic to the 2-torus.

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Grünbaum grunbaum03
  • Theorem 2.9: Grünbaum grunbaum03
  • Theorem 2.10: Grünbaum grunbaum03
  • Definition 2.11
  • Definition 2.12
  • Definition 2.13
  • ...and 37 more