$2$-complexes with unique embeddings in 3-space
Agelos Georgakopoulos, Jaehoon Kim
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.