Embedding simply connected 2-complexes in 3-space

TL;DR

This paper characterizes when simply connected 2-complexes can be embedded in 3-space using nine excluded minors, extending classical planarity results to higher-dimensional complexes.

Contribution

It provides a Kuratowski-type characterization for embeddability of simply connected 2-complexes in 3-space, answering longstanding open questions.

Findings

Nine excluded minors determine embeddability.

Characterization applies to all simply connected 2-complexes.

Extension of classical planarity to higher dimensions.

Abstract

Firstly, we characterise the embeddability of simply connected locally 3-connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski's characterisation of graph planarity, by nine excluded minors. This answers questions of Lov\'asz, Pardon and U. Wagner. The excluded minors are the cones over $K_5$ and $K_{3,3}$, five related constructions, and the remaining two are obtained from triangulations of the M\"obius strip by attaching a disc at its central cycle. Secondly, we extend the above theorem to all simply connected 2-dimensional simplicial complexes.