Timeline for Are loop spaces of homotopically equivalent spaces homotopically equivalent?
Current License: CC BY-SA 3.0
7 events
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| Mar 11, 2013 at 11:07 | comment | added | Jacob Bell | @ricardo: I agree, I put it out there as, to me, this perspective sweeps under the rug the actual maths and makes things look formal and easy. I did say in my second comment that I was cheating. :) | |
| Mar 11, 2013 at 1:34 | comment | added | Ricardo Andrade | This is a fairly overpowered answer to a rather simple question... By the way, the homotopy invariance (with respect to homotopy equivalences) of homotopy pullbacks holds for all spaces (not just "nice" spaces), and is a fairly elementary exercise. Even more generally, homotopy limits of spaces preserve homotopy equivalences. This is a consequence of the homotopy invariance of homotopy limits in model categories applied to the Strøm model structure (see ncatlab.org/nlab/show/Strom+model+structure), together with the fact that all objects are fibrant in the Strøm model structure. | |
| Mar 10, 2013 at 19:13 | comment | added | Jacob Bell | but definitely understanding why the homotopy fibre product $\bullet \times_X^h \bullet$ is the (based) loopspace is worth the time (~30 seconds). It's such a cool thing! | |
| Mar 10, 2013 at 19:12 | comment | added | Aaron Mazel-Gee | CW complexes are the opposite of "pathological". | |
| Mar 10, 2013 at 19:12 | comment | added | Jacob Bell | for CW complexes everything is fine. But I sympathise if you think my answer is basically cheating. I'm not a topologist myself, but I got used to some homotopy ideas and now I just take them for granted. | |
| Mar 10, 2013 at 18:56 | comment | added | Hiro | Thanks for your comment. I do not know anything about homotopy limits, but how about if the spaces are CW complexes? | |
| Mar 10, 2013 at 18:11 | history | answered | Jacob Bell | CC BY-SA 3.0 |