TITLE: Example of a CW complex not homeomorphic to the realization of a simplicial set?
QUESTION [11 upvotes]: 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)?

REPLY [10 votes]: The geometric realization of a simplicial set is always triangulable. See Corollary 4.6.12 in Cellular Structures in Topology 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). This paper contains another example and shows, on the other hand, that every CW-complex with cells in at most two dimensions is triangulable.