Timeline for Symmetric line spaces are homeomorphic to Euclidean spaces
Current License: CC BY-SA 4.0
15 events
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| Apr 2, 2023 at 4:19 | comment | added | Moishe Kohan | @TarasBanakh: I see. I did not know this theorem, my knowledge of topological transformation groups is limited to the results before 1960s. | |
| Mar 30, 2023 at 5:25 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Mar 29, 2023 at 19:30 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Mar 29, 2023 at 17:10 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added a solution of the Conjecture
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| Mar 29, 2023 at 17:01 | comment | added | Taras Banakh | @MoisheKohan By the assumption, the space is isometrically homogeneous and bounded subsets are compact. Those assumptions imply that its isometry group is locally compact. So the space is a quotient space of a locally compact group, and being locally contractible, is a topological manifold, by a result of Szenthe or Hoffmann or Antonyan (there were three different proofs of the same result of Szenthe whose proof contained a gap). And topological manifolds are finite-dimensional. Of course, this fact is neither obvious nor trivial, but true nonetheless. | |
| Mar 29, 2023 at 13:47 | comment | added | Moishe Kohan | It is still unclear to me how to get finite dimension (I am worried about some versions of the Hilbert-cube-manifolds). I do not see how one can use Cartan in your setting since he only classifies Riemannian symmetric spaces. | |
| Mar 29, 2023 at 8:42 | history | edited | Taras Banakh | CC BY-SA 4.0 |
edited body
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| Mar 29, 2023 at 8:33 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Rewrote the problem since the notion of a Busemann G-space is more general than my linear metric spaces.
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| Mar 29, 2023 at 4:28 | history | edited | Taras Banakh |
edited tags; edited tags
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| Mar 29, 2023 at 3:33 | comment | added | Taras Banakh | @MoisheKohan As far as I can understand, Montgomery-Zippin can help in proving that the symmetric Busemann G-space is a homogeneous space $G/H$ of a Lie group $G$, so has a structure of a smooth manifold. But how to prove (from Montgometry-Zippin) that it is homeomorphic to an Euclidean space? Exactly for this task I had an idea to apply Cartan's classification of symmetric spaces. Finite dimension should follow from the compactness of closed balls in the symmetry of a Busemann G-space, namely, from the local compactness of its isometry group. | |
| Mar 29, 2023 at 3:27 | comment | added | Taras Banakh | @BenoîtKloeckner Thank you for the comment. Indeed, I forgot to add this equality, which is added now. | |
| Mar 29, 2023 at 3:26 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Mar 28, 2023 at 21:31 | comment | added | Moishe Kohan | I do not think Cartan is relevant here, but Montgomery-Zippin et al should do the job. But maybe you'll have to add the assumption of finite dimension. | |
| Mar 28, 2023 at 21:16 | comment | added | Benoît Kloeckner | I am confused by the definition. Certainly uniqueness in "medial" points out to a more restrictive definition of midpoints (adding $d(x,y)=d(y,z)$ maybe?), right? | |
| Mar 28, 2023 at 19:08 | history | asked | Taras Banakh | CC BY-SA 4.0 |