Timeline for Are there topological obstructions to the existence of almost quaternionic structures on compact manifolds?
Current License: CC BY-SA 4.0
17 events
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| Dec 3, 2022 at 17:17 | history | edited | LSpice | CC BY-SA 4.0 |
`\DeclareMathOperator`, and links to answers, while this is on the front page; restriction -> obstruction
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| Mar 22, 2014 at 19:25 | history | edited | user9072 |
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| Jun 5, 2012 at 17:21 | answer | added | Oldřich Spáčil | timeline score: 9 | |
| Feb 26, 2011 at 17:35 | vote | accept | Andrei Moroianu | ||
| Feb 8, 2011 at 10:13 | history | edited | Andrei Moroianu | CC BY-SA 2.5 |
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| Jan 20, 2011 at 18:38 | history | edited | Andrei Moroianu | CC BY-SA 2.5 |
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| Jan 20, 2011 at 18:24 | comment | added | Andrei Moroianu | @JME: I think you misunderstand the notion of almost quaternionic structure. From a logical point of view, almost quaternionic is to quaternion-K\"ahler EXACTLY what almost complex is to K\"ahler. | |
| Jan 20, 2011 at 8:29 | comment | added | Andrei Moroianu | @JME: I try to understand your comment. For me, almost quaternionic is clearly equivalent to the existence of structure reduction to $Sp(n)Sp(1)$. May be you mean some integrability condition: Joyce could have constructed almost quaternionic manifolds which are not quaternion-K\"ahler? Anyway, do you have a reference for Joyce's paper? | |
| Jan 19, 2011 at 12:52 | answer | added | Andrei Moroianu | timeline score: 7 | |
| Jan 18, 2011 at 21:33 | answer | added | Thomas Kragh | timeline score: 6 | |
| Jan 18, 2011 at 19:35 | comment | added | Andrei Moroianu | @Thomas: this is standard notation,, although, of course slightly misleading. In fact $Sp(1)$ is obtained by right multiplication with unit quaternions on $\mathbb{H}^n$, while $Sp(n)$ is the centralizer of $Sp(1)$, and is given by left multiplication with matrices with quaternionic entries. The diagonal $Sp(1)\subset Sp(n)$ is of course different from the former $Sp(1)$! | |
| Jan 18, 2011 at 19:28 | comment | added | Spiro Karigiannis | $Sp(1) Sp(n)$ is shorthand notation in this context denoting the Lie group $Sp(1) \times Sp(n) / \{\pm 1\}$. | |
| Jan 18, 2011 at 19:20 | comment | added | Thomas Kragh | What do you mean by $Sp(1)Sp(n)$? since $Sp(1)$ is a sub-Lie-group of $Sp(n)$ this is with the obvious definition simply $Sp(n)$ gain. | |
| Jan 18, 2011 at 19:12 | history | edited | Andrei Moroianu | CC BY-SA 2.5 |
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| Jan 18, 2011 at 12:16 | comment | added | Andrei Moroianu | No, this is the point: almost quaternionic does not imply almost complex (see the example of S^4). More generally, the quaternionic projective spaces $\mathbb{H}\mathrm{P}^n$ are quaternion-K\"ahler, but have no almost complex structure (Hirzebruch, 1953). | |
| Jan 18, 2011 at 12:07 | comment | added | Gunnar Þór Magnússon | Isn't the case of $S^8$ (or $S^{2n}$ for $n \geq 4$) also excluded since they don't admit almost complex structures? | |
| Jan 18, 2011 at 10:20 | history | asked | Andrei Moroianu | CC BY-SA 2.5 |