Timeline for Are there CW structures on homotopy limits of CW maps?
Current License: CC BY-SA 3.0
7 events
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| Jul 24, 2013 at 19:07 | comment | added | Eric Wofsey | Do you really mean to ask whether there is a factorization of $g$ as $X\to Z\to Y$ where the first map is a homotopy equivalence, the second map is a fibration, and $Z$ is a CW complex? If by "fibration" you mean Serre fibration, the answer is yes (by the small object argument construction of the factorization of $g$ into an acyclic cofibration and a fibration). If you want to demand a Hurewicz fibration, the problem seems harder. | |
| Jul 24, 2013 at 18:24 | comment | added | Vidit Nanda | @EricWofsey yes the path space is awful, but here's a question: do we really need the entire path space to build the homotopy limit, or does it suffice (say) to only consider homotopy classes of paths? | |
| Jul 24, 2013 at 17:37 | comment | added | Eric Wofsey | I don't know how to give a rigorous proof, but I believe $L_H$ should almost never be a CW complex--it's infinite-dimensional, in a way that's like a Banach space (essentially, you're looking at a space of functions with the sup metric) rather than like a countably infinite-dimensional vector space that can be topologized as the colimit of its finite dimensional subspaces. However, $L_H$ does always have the homotopy type of a CW complex. Milnor proved a very general result along these lines in his paper "On spaces having the homotopy type of a CW-complex". | |
| Jul 24, 2013 at 16:57 | comment | added | Vidit Nanda | @RicardoAndrade $f$ and $g$ are to be treated as arbitrary cell maps. It doesn't help too much to treat $g$ as the identity on $Y$: basically, if we could put a CW structure on $Y^I$ itself, then immediately we would get $L_H$ as a subcomplex of $X \times Y^I$ for any cell map $g$. | |
| Jul 24, 2013 at 16:53 | history | edited | Vidit Nanda | CC BY-SA 3.0 |
edited to account for ricardo's comments
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| Jul 24, 2013 at 16:24 | comment | added | Ricardo Andrade | Here are a few simple remarks. Regarding the second sentence, you probably need $f$ to be a cellular map in order to get a CW-structure on its mapping cylinder. Otherwise, you probably only get a cell structure. Further, do you allow for non-trivial conditions to be imposed on the function $g$? If not, then taking $g$ to be the identity on $Y$ we get $L_H = Y^I$, which slightly reduces our task. | |
| Jul 24, 2013 at 13:15 | history | asked | Vidit Nanda | CC BY-SA 3.0 |