Timeline for When is the quotient by an $n$-fold loop space an $m$-fold loop space?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Sep 10, 2015 at 23:21 | vote | accept | Jonathan Beardsley | ||
| Sep 10, 2015 at 21:24 | answer | added | MatanP | timeline score: 6 | |
| Sep 10, 2015 at 18:10 | vote | accept | Jonathan Beardsley | ||
| Sep 10, 2015 at 23:21 | |||||
| Sep 10, 2015 at 16:56 | answer | added | Qiaochu Yuan | timeline score: 5 | |
| Sep 9, 2015 at 19:54 | history | edited | Jonathan Beardsley | CC BY-SA 3.0 |
added 1590 characters in body
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| Sep 9, 2015 at 19:25 | comment | added | Jonathan Beardsley | Admittedly, I now see that the homotopy cofiber is the wrong thing to be doing here. | |
| Sep 9, 2015 at 15:52 | answer | added | Ilias A. | timeline score: 7 | |
| Sep 9, 2015 at 13:40 | comment | added | Jonathan Beardsley | @ChrisSchommer-Pries perhaps I'm doing something stupid here, but when I have a fibration, say, e.g. $\Omega SU(n)\to \Omega SU(n+1)\to \Omega S^{2n+1}$, everything here is a loop space and a loop map (it's just loops on a fibration). But at the same time $\Omega S^{2n+1}$ is a sort of quotient group, so I feel that it should have some kind of quotient property, but perhaps I'm being really dumb here... | |
| Sep 9, 2015 at 10:07 | answer | added | Jesper Grodal | timeline score: 11 | |
| Sep 9, 2015 at 7:18 | comment | added | Chris Schommer-Pries | Another example is $S^1 \to S^1$ via the double covering map. $S^1 = K(\mathbb{Z},1)$ is an infinite loop space and this is an infinite loop map, but the cofiber is $\mathbb{RP}^2$. I don't think that is even a 1-fold loop space. Do you have any examples of this happening where $n>1$, $m>0$, and the quotient map $Y \to Y/X$ is an m-fold loop map? (presumably you meant for this last condition in your question, otherwise $Y/X$ could just accidentally happen to have an m-fold loop space structure with nothing to do with $X$ or $Y$). | |
| Sep 9, 2015 at 4:33 | comment | added | Eric Wofsey | When you say "$n$-fold loop space", do you mean that literally or only up to homotopy equivalence? I would have assumed the latter, except that if so I don't know why you tagged the question "general topology"... | |
| Sep 9, 2015 at 4:09 | comment | added | Qiaochu Yuan | @Prasit: no. For example, take $f : X \to Y$ to be an inclusion of discrete groups (so $n = 1$). The homotopy cofiber of $f$ is $Y$ with the image of $X$ identified to a point, which is very different from the group-theoretic quotient $Y/X$ (which still need not be a group of $X$ isn't normal). That is, there's a forgetful functor from $n$-fold loop spaces to spaces, and it's not even close to preserving homotopy cofibers. | |
| Sep 9, 2015 at 4:03 | comment | added | Prasit | What if the map $f:X \to Y$ is a m-fold loop map, i.e. $f = \Omega^m g$ for some $g$ and $m\leq n$? Isn't then $Y/X$ a $m$-fold loop space? | |
| Sep 9, 2015 at 3:30 | history | asked | Jonathan Beardsley | CC BY-SA 3.0 |