Timeline for Is a quotient of real linear algebraic groups always a Cartesian product of compact and contractible factors?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 2, 2022 at 14:33 | comment | added | Ian Gershon Teixeira | What a great result! Would you happen to have any references for that or a proof sketch? I would love it if you could answer the question your example about the tangent bundle of the sphere inspired me to ask here: mathoverflow.net/questions/412741/… | |
| Jan 2, 2022 at 7:52 | comment | added | Nicolast | Yes, sorry, $G_{\mathbb C}/H_{\mathbb C}$ is the tangent bundle to $G_{\mathbb R}/H_{\mathbb R}$ where $G_{\mathbb R}$ and $H_{\mathbb R}$ are the compact real forms of $G_{\mathbb C}$ and $H_{\mathbb C}$. | |
| Jan 1, 2022 at 21:26 | comment | added | Ian Gershon Teixeira | Hmm but what about when $ G=SL_2 $ and H is trivial? $ SL_2(\mathbb{C}) $ is homotopy equivalent to $ SU_2 $ but the tangent bundle of $ SL_2(\mathbb{R}) $ is homotopy equivalent to $ SO_2(\mathbb{R}) $ so they can't be diffeomorphic. Maybe it's true as long as the real points of $ G $ and $ H $ are both compact? | |
| Jan 1, 2022 at 13:18 | comment | added | Nicolast | I think it is always true, at least when $G$ and $H$ are reductive. | |
| Dec 29, 2021 at 13:54 | comment | added | Ian Gershon Teixeira | Is it true in general that as a real smooth manifold $ G_{\mathbb{C}}/H_{\mathbb{C}} $ is always the tangent bundle of $ G_{\mathbb{R}}/H_{\mathbb{R}} $ or is there something special about this case with the sphere? | |
| Dec 10, 2021 at 20:27 | comment | added | Nicolast | yes exactly, you'd get the tangent bundle to $P^n_{\mathbb R}$ which is also non-trivial unless $n\in \{1,3,7\}$ (since the sphere is its double cover). | |
| Dec 10, 2021 at 16:20 | comment | added | Ian Gershon Teixeira | So (just to make sure I'm understanding you correctly) in the $ n=2 $ case this would be the sub variety of $ \mathbb{C}^3 $ defined by $ x^2+y^2+z^2=1 $? Also what would happen if we took $ G= O_{n+1}(\mathbb{C}) $ and $ H=O_n(\mathbb{C}) \times O_1(\mathbb{C}) $ ? I think we get the normal bundle over real projective space $ P_\mathbb{R}^n $. Is that bundle also nontrivial? | |
| Dec 10, 2021 at 15:47 | vote | accept | Ian Gershon Teixeira | ||
| Dec 10, 2021 at 15:18 | history | answered | Nicolast | CC BY-SA 4.0 |