Timeline for Is every manifold a double coset space?
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Dec 9, 2019 at 10:50 | comment | added | YCor | I'd be curious about the connected sum of two (or more) copies of $S^1\times S^2$ (or 3-tori). | |
| Dec 9, 2019 at 3:29 | comment | added | Ryan Budney | I have a vague recollection that the answer is negative, and that this is an old theorem of Wu-Chung Hsiang's. I believe he shows exotic spheres are not of this form. | |
| Dec 9, 2019 at 1:47 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Dec 9, 2019 at 1:42 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Nov 27, 2019 at 18:45 | comment | added | Ryan Budney | I can't speak for Kapovich but my assumptions would lead me to think he is referring to the type of double-quotient construction used to make geometric manifolds. | |
| Nov 27, 2019 at 15:58 | comment | added | Robbie Lyman | it is true for all 2-manifolds, (I think this requires assuming that your definition of manifold requires second countability) in the sense that noncompact 2-manifolds with nontrivial fundamental group can be shown to have the hyperbolic plane as their universal cover. | |
| Nov 27, 2019 at 14:23 | comment | added | Ian Gershon Teixeira | @FrancoisZiegler The commit about “a bit more general” is leftover from before the first edit when the question was more general. I’ve deleted it. Now everything should match the title. | |
| Nov 27, 2019 at 14:22 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Nov 27, 2019 at 14:21 | comment | added | Ian Gershon Teixeira | @Will Sawin I know all surfaces (both orientable and non orientable) arise this way, that’s part of why I asked the question. Is it true for all connected 2 manifolds (not necessarily compact)? | |
| Nov 27, 2019 at 5:52 | comment | added | Francois Ziegler | @YCor Well, biquotients appear to have had several definitions, and then the OP seems to want “a bit more general”. (How exactly?) | |
| Nov 27, 2019 at 5:21 | comment | added | YCor | @FrancoisZiegler I understand the current question with this specific way (although I'd have denoted $\Gamma\backslash G/H$) | |
| Nov 27, 2019 at 4:30 | comment | added | Francois Ziegler | The question still doesn’t match the title. (Double coset space entails a specific way for $\Gamma$ to act on $G/H$.) | |
| Nov 27, 2019 at 3:39 | comment | added | Will Sawin | The fact that all surfaces arise this way should be noted. | |
| Nov 27, 2019 at 3:00 | comment | added | Ian Gershon Teixeira | Similarly I'd be interested in a large class of manifolds all of which come from homogeneous spaces by quotienting out by a group action. I'd be interested in results of the sort "manifolds with X geometric structure always arise as the quotient by a free and proper action of their fundamental group on their universal cover and their universal cover is always homogeneous." Again I should probably make another question. | |
| Nov 27, 2019 at 2:56 | comment | added | Ian Gershon Teixeira | @Ycor Ok you are right. I edited the question so it aligns with the title. I expect the answer now to be "no". The question I'm more interested in is if $ \Gamma $ is not necessarily a subgroup. In particular I'm curious if there is a general enough structure such that every manifold "come from a lie group/ comes from a homogeneous space" using only algebraic data (e.g. quotienting by group actions). Although I should probably just ask that in a different question with a different title. | |
| Nov 27, 2019 at 2:52 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Nov 26, 2019 at 5:51 | comment | added | YCor | The question as currently stated ($\Gamma$ is not assumed to lie in $G$) doesn't match the title. | |
| Nov 26, 2019 at 5:17 | history | asked | Ian Gershon Teixeira | CC BY-SA 4.0 |