When can a contractible 2-complex be embedded in R^3? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:51:58Z http://mathoverflow.net/feeds/question/72839 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72839/when-can-a-contractible-2-complex-be-embedded-in-r3 When can a contractible 2-complex be embedded in R^3? unknown (google) 2011-08-13T18:50:02Z 2011-08-13T21:09:37Z <p>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?</p> http://mathoverflow.net/questions/72839/when-can-a-contractible-2-complex-be-embedded-in-r3/72847#72847 Answer by Agol for When can a contractible 2-complex be embedded in R^3? Agol 2011-08-13T21:09:37Z 2011-08-13T21:09:37Z <p>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. </p>