Dual matroids of 2-complexes -- revisited

TL;DR

This paper establishes a criterion linking the embeddability of simply connected local 2-complexes in 3-space to their dual matroids being graphic, extending Whitney's planarity criterion to three dimensions.

Contribution

It introduces a new characterization of 2-complex embedding in 3-space via dual matroids, highlighting the necessity of certain topological assumptions.

Findings

Simply connected local 2-complexes embed in 3-space iff their dual matroids are graphic

Examples show the assumptions of simply connectedness and locality are necessary

Extends Whitney's planarity criterion to 3-dimensional complexes

Abstract

We prove that simply connected local 2-dimensional simplicial complexes embed in 3-space if and only if their dual matroids are graphic. Examples are provided that the assumptions of simply connectedness and locality are necessary. This may be regarded as a 3-dimensional analogue of Whitney's planarity criterion from 1932.