Timeline for Is there a spherical analogue of polar duality for spherical complexes?
Current License: CC BY-SA 4.0
19 events
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| May 11, 2020 at 15:17 | comment | added | Sam Hopkins | The spherical complexes you define should be a very special class of PL decompositions of the sphere. I would recommend looking at that part of "Oriented matroids" to understand the definitions more precisely. | |
| May 11, 2020 at 14:56 | comment | added | Malkoun | Yes, but I was wondering essentially about the last sentence in your answer, when you wrote "it is not clear that the dual PL cell decomposition will correspond to a polyhedral fan". Can you explain a bit more please this statement? | |
| May 11, 2020 at 13:43 | comment | added | Sam Hopkins | I think everything I’m talking about are CW complexes (maybe even regular CW complexes). You’d have to look at the references to be sure. | |
| May 11, 2020 at 9:23 | comment | added | Malkoun | I never studied PL cell decompositions. Just out of curiosity, is there a PL cell decomposition of a sphere (represented as a higher dimensional cube), which is not equivalent as a CW complex to that of a spherical complex? I am not sure if my question makes sense (I don't know the definitions well). | |
| May 10, 2020 at 21:19 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| May 10, 2020 at 20:47 | comment | added | Malkoun | Yes, but this is good for me, because the fan I am interested in is a normal fan. However, it would be interesting to find a more general and more direct construction, should there exist one. | |
| May 10, 2020 at 20:47 | comment | added | Sam Hopkins | (But that case is probably not so interesting for you, because I think it means you could deform your spherical complex into a convex polytope.) | |
| May 10, 2020 at 20:46 | comment | added | Sam Hopkins | Yes of course: in the polytopal case things work perfectly. | |
| May 10, 2020 at 20:45 | comment | added | Malkoun | By the way, in case the complete polyhedral fan $T$ is the normal fan of a convex polytope $P$, one may view the normal fan of $P^*$, with $P^*$ being the polar dual of $P$, as a kind of dual of $T$. So this seems to provide a partial answer to my question in this important special case. | |
| May 10, 2020 at 19:49 | comment | added | Sam Hopkins | At any rate you should unaccept this answer, because while it might be useful for you as a starting point, it doesn't answer your question. I'm now convinced your question is very interesting, though, and I'm surprised I've never heard discussion of "dual polyhedral fans" before. One possible answer is that only genuinely polytopal fans have duals: an analogous thing is that only planar graphs have duals. | |
| May 10, 2020 at 19:35 | comment | added | Sam Hopkins | You can maybe mimic the construction of a normal cone/normal fan, but for the faces of your spherical complex instead of a convex polytope: en.wikipedia.org/wiki/Normal_fan. | |
| May 10, 2020 at 19:32 | comment | added | Malkoun | I understand your objection. Can one perhaps construct a fan out of the duals of the polyhedral cones? | |
| May 10, 2020 at 19:30 | comment | added | Sam Hopkins | What you kinda want to do is take the normal fan of a fan. But I'm not sure that makes sense. | |
| May 10, 2020 at 19:27 | comment | added | Sam Hopkins | Even the most basic example of say 6 equiangular cones in $\mathbb{R}^2$. These cones are acute so their duals will be too big. | |
| May 10, 2020 at 19:26 | comment | added | Malkoun | Could you give an example where the dual cones of a fan do not fit together into a fan please? | |
| May 10, 2020 at 19:25 | comment | added | Sam Hopkins | No, what's troubling me is that if I take all the cones in a fan and take their duals, they will not fit together into a fan. | |
| May 10, 2020 at 19:20 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| May 10, 2020 at 18:21 | vote | accept | Malkoun | ||
| May 10, 2020 at 19:49 | |||||
| May 10, 2020 at 17:42 | history | answered | Sam Hopkins | CC BY-SA 4.0 |