2-complexes with unique embeddings in 3-space

2021/09/09 by Agelos Georgakopoulos, Jaehoon Kim, Georgakopoulos, Agelos +1
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Topological and Geometric Data Analysis #math.CO #math.GT

paper · pdf · doi:10.48550/arxiv.2109.04085

arxiv created 2021/09/09 · openalex publication_date 2021/09/09 · arxiv updated 2021/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected 2-complex every link graph of which is 3-connected admits an essentially unique locally flat embedding into the 3-sphere, if it admits one at all. This can be thought of as a generalisation of the 3-dimensional Schoenflies theorem.

Related

• A simple and elementary proof of Whitney's unique embedding theorem
• Embedding simply connected 2-complexes in 3-space -- II. Rotation systems
• Embedding simply connected 2-complexes in 3-space -- V. A refined Kuratowski-type characterisation
• An obstruction to embedding 2-dimensional complexes into the 3-sphere
• Embedding simply connected 2-complexes in 3-space -- I. A Kuratowski-type characterisation