Let $X$ be a contractible 2-dimensional simplicial complex. Are there nice necessary and sufficient conditions for $X$ to be embeddable in $\mathbb R^3$? Clearly it is necessary that the link of every vertex be a planar graph. Is this sufficient?



When can a contractible 2-complex be embedded in R^3?
If your complex is finite, then figure out the possible ways of 
thickening it to a 3-manifold. The possible thickenings are
determined by the various embeddings of the links of the vertices
into $S^2$, then seeing if these induce compatible thickenings
over the edges (determined by the same cyclic ordering over
the link of the edge) and faces of the complex. If it can be thickened
this way, then it must be a ball since it is a contractible 3-manifold. 


