$2$-complexes with unique embeddings in 3-space

TL;DR

This paper extends Whitney's theorem to 3-dimensional complexes, proving that certain simply-connected 2-complexes have essentially unique embeddings into the 3-sphere, generalizing the Schoenflies theorem.

Contribution

It establishes a 3D analogue of Whitney's theorem, showing uniqueness of embeddings for a class of 2-complexes in 3-space.

Findings

Unique embedding for certain 2-complexes in 3-sphere

Generalization of the 3D Schoenflies theorem

Conditions on link graphs for embedding uniqueness

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.