From planar graph layout to flat embeddings of 2-complexes

A question about 2-complexes, analogous to graph planarity, is whether they can be embedded in 3 dimensions. We consider three methods for straight-edge planar graph layout: barycentric maps (and convex-combination maps) which are known to fail in 3 dimensions, Read's vertex-deletion method, and the De Fraysseix-Pach-Pollack method, and give more examples showing that each method can fail.