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)?
2 Answers
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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3$\begingroup$ 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. $\endgroup$ Jul 16, 2013 at 13:43
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2$\begingroup$ Oh. That was an answer to the other part of the question. $\endgroup$ Jun 3, 2017 at 14:27
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1$\begingroup$ I could have said the nerve of a nontrivial group. $\endgroup$ Jun 3, 2017 at 19:53
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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$\begingroup$ The link to the paper at
springerlink.com
is broken. I'm also unable to find any copy saved on the Wayback Machine. $\endgroup$ Aug 24, 2022 at 5:45