Let $X$ be a contractible 2dimensional 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?

$\begingroup$ Its not necessary at all  consider the space that's the product of an interval with the cone on a discrete space (say, with at least three elements). This embeds in $\mathbb R^3$ and is contractible, but does not satisfy your "necessary" condition. $\endgroup$– Ryan BudneyAug 13 '11 at 20:04

2$\begingroup$ @Ryan, it seems to me that no matter how you triangulate your example, all the links are suspensions of discrete spaces, which are planar graphs. $\endgroup$– Autumn KentAug 13 '11 at 20:31

$\begingroup$ (I guess at some points the links are cones on discrete spaces, but that's okay, too.) $\endgroup$– Autumn KentAug 13 '11 at 20:32

1$\begingroup$ Er, right. I had some examples in mind. Maybe they demonstrated something else. For example, think about the bundle over $S^1$ whose fibre is the cone on a threepoint space, so the fibre is a "Y". If the monodromy of that bundle flips the two "arms" of the "Y", but keeps the base fixed, the total space would be a Moebius band with an annulus attached along the central circle. Give it a triangulation which is a refinement of the product CWdecomposition. The links of vertices are planar graphs but I believe it can't embed since if it did, the Moebius band would be orientable. $\endgroup$– Ryan BudneyAug 13 '11 at 22:33

$\begingroup$ So what this demonstrates is that the author's condition is very much just a local one. Perhaps you could view it as the 0dimensional criterion in an obstruction theoretic approach. The above example demonstrates you need some kind of 1dimensional obstruction, as well. $\endgroup$– Ryan BudneyAug 13 '11 at 22:36
If your complex is finite, then figure out the possible ways of thickening it to a 3manifold. 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 3manifold.

$\begingroup$ Very nice answre! This still leaves open the question of whether there exist 2dimensional complexes all of whose links are planar, but that don't embed in $\mathbb R^3$. $\endgroup$ Aug 13 '11 at 22:15

$\begingroup$ @Andre: I believe my 2nd example in the comments above address your question. $\endgroup$ Aug 13 '11 at 22:37

7$\begingroup$ @Ryan: Your Yshaped Moebius band example is not contractible. So, strictly speaking, it doesn't answer the question. However, a small variation does: replace "the total space would be a Moebius band with an annulus attached along the central circle" by "the total space would be a Moebius band with A DISK attached along the central circle". $\endgroup$ Aug 13 '11 at 22:48

$\begingroup$ Ah, yes! My mind is all over the place today. $\endgroup$ Aug 13 '11 at 22:51