A simplicial complex which is not collapsible, but whose barycentric subdivision is - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:47:59Z http://mathoverflow.net/feeds/question/87557 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87557/a-simplicial-complex-which-is-not-collapsible-but-whose-barycentric-subdivision A simplicial complex which is not collapsible, but whose barycentric subdivision is Andy Soffer 2012-02-05T02:33:50Z 2012-05-15T00:02:37Z <p>Does anyone know of a simplicial complex which is not collapsible but whose barycentric subdivision is?</p> <p>Every collapsible complex is necessarily contractible, and subdivision preserves the topological structure, so we are certainly looking for a complex which is contractible, but not collapsible. The only complexes I know of which are contractible but not collapsible are the dunce cap and Bing's house with two rooms. Neither of these have any free faces, and so no iterated subdivision will result in a collapsible complex.</p> http://mathoverflow.net/questions/87557/a-simplicial-complex-which-is-not-collapsible-but-whose-barycentric-subdivision/87666#87666 Answer by Sergey Melikhov for A simplicial complex which is not collapsible, but whose barycentric subdivision is Sergey Melikhov 2012-02-06T14:09:00Z 2012-05-15T00:02:37Z <p><a href="http://www.ams.org/journals/tran/1969-137-00/S0002-9947-1969-0238288-X/S0002-9947-1969-0238288-X.pdf" rel="nofollow">Lickorish and Martin</a> constructed, for each \$r\$, a triangulation of the \$3\$-ball whose \$r\$th barycentric subdivision collapses, but \$(r-1)\$th doesn't. The basic idea, going back to <a href="http://infoshako.sk.tsukuba.ac.jp/~HACHI/math/library/knot_eng.html" rel="nofollow">Furch</a> and Bing, is to triangulate a cube with a knotted hole, where the missing knot has bridge index \$2^r+1\$, and then fill back a small part of the hole - so that topologically, no hole remains, but the \$3\$-ball now contains a knot triangulated by a single edge.</p> <p><strong>Added later:</strong> <a href="http://www.ams.org/journals/tran/1972-170-00/S0002-9947-1972-0310899-2/" rel="nofollow">Kearton and Lickorish</a> also constructed triangulations of the \$n\$-ball, \$n\ge 3\$, whose \$r\$th barycentric subdivision is not collapsible. On the other hand, every triangulation of a ball becomes collapsible after some number of barycentric subdivisions, according to a recent preprint by <a href="http://arxiv.org/abs/1202.6606" rel="nofollow">Adiprassito and Benedetti</a> (see their Corollary 3.5).</p>