Timeline for real and complex vector spaces as topological categories
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 24, 2016 at 8:56 | comment | added | Yonatan Harpaz | Right! Silly of me. I added an edit to correct it. By the way, the argument above seems to work quite generally. If $f: C \rightleftarrows D: g$ is an adjunction of $\infty$-categories then the induced map $g_*:Aut^0(X) \to Aut^0(Y)$ is a retract inclusion if $X$ is in the essential image of $f$, and vice-versa. | |
| Feb 24, 2016 at 8:49 | history | edited | Yonatan Harpaz | CC BY-SA 3.0 |
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| Feb 24, 2016 at 8:27 | comment | added | KotelKanim | NIce! thanks for the answer Yonatan, I kind of suspected the answer is no, and it is good to see a proof (a small nitpick, the group $GL(U(\mathbb{C}^n))$ is not connected, but you can restrict to $SL$ and continue the argument without change). | |
| Feb 24, 2016 at 8:08 | vote | accept | KotelKanim | ||
| Feb 23, 2016 at 19:31 | comment | added | Yonatan Harpaz | Yes, I believe you're right. Indeed, $\pi_1(O(4)) = \pi_1(O(3))$ is already $\mathbb{Z}/2$. | |
| Feb 23, 2016 at 16:07 | comment | added | Denis Nardin | At the limit the map $U\to O$ is certainly not a retract because it is not injective on homotopy groups (in particular on $\pi_1$, where $\pi_1U=\mathbb{Z}$ and $\pi_1O=\mathbb{Z}/2$). So for $n\gg0$ the map $U_n\to O_{2n}$ cannot be a retraction. In fact I think this proves it even for $n=2$ | |
| Feb 23, 2016 at 15:50 | history | edited | Yonatan Harpaz | CC BY-SA 3.0 |
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| Feb 23, 2016 at 13:40 | history | answered | Yonatan Harpaz | CC BY-SA 3.0 |