Example of a CW complex not homeomorphic to the realization of a simplicial set? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:12:19Z http://mathoverflow.net/feeds/question/34674 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34674/example-of-a-cw-complex-not-homeomorphic-to-the-realization-of-a-simplicial-set Example of a CW complex not homeomorphic to the realization of a simplicial set? Harry Gindi 2010-08-05T19:14:16Z 2010-08-05T20:18:15Z <p>I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't isomorphic to the total singular complex of a CGWH space. Are there?) Would someone mind providing an example of one (and an example for the opposite statement as well, if it is true)?</p> http://mathoverflow.net/questions/34674/example-of-a-cw-complex-not-homeomorphic-to-the-realization-of-a-simplicial-set/34677#34677 Answer by Tom Goodwillie for Example of a CW complex not homeomorphic to the realization of a simplicial set? Tom Goodwillie 2010-08-05T19:59:46Z 2010-08-05T19:59:46Z <p>The mapping cylinder of a really messy continuous map $I\to I$</p> <p>The nerve of the category in which there are two objects and each Hom set is a singleton.</p> http://mathoverflow.net/questions/34674/example-of-a-cw-complex-not-homeomorphic-to-the-realization-of-a-simplicial-set/34679#34679 Answer by Evan Jenkins for Example of a CW complex not homeomorphic to the realization of a simplicial set? Evan Jenkins 2010-08-05T20:18:15Z 2010-08-05T20:18:15Z <p>The geometric realization of a simplicial set is always triangulable. See Corollary 4.6.12 in <a href="http://books.google.com/books?id=Vg7Z75qqmDIC" rel="nofollow"><i>Cellular Structures in Topology</i></a> by Fritsch and Piccinini. They also give an explicit example (in section 3.4) of a non-triangulable CW-complex (which uses, I think, essentially the same idea as Tom Goodwillie's suggestion). <a href="http://www.springerlink.com/content/u31765707336555u/" rel="nofollow">This paper</a> contains another example and shows, on the other hand, that every CW-complex with cells in at most two dimensions is triangulable.</p>