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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)?

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Heh, I just realized that we could call the "opposite statement" the "adjoint statement". – Harry Gindi Aug 5 '10 at 20:13
up vote 9 down vote accepted

The mapping cylinder of a really messy continuous map $I\to I$

The nerve of the category in which there are two objects and each Hom set is a singleton.

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What do you mean by the nerve in the second example? I thought the nerve was a simplicial set by definition, so it can't possibly provide an example of a CW-complex that doesn't come from a simplicial set. – Omar Antolín-Camarena Jul 16 '13 at 13:43

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.

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