Timeline for Topologies on diffeomorphisms groups
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 13, 2023 at 13:34 | comment | added | HUO | Thanks. Why not posting this as an answer? | |
| Nov 13, 2023 at 12:36 | comment | added | YCor | No. Write $M$ as increasing union of compact subsets $K_n$ so that the interiors cover too. Let $G_n$ be the subgroup of diffeos supported in $K_n$. It's closed under the given convergence, so for a metrizable topology with such converging sequences, is closed. It has empty interior. For a topological group, it just means it's not open. Indeed, just approximate the identity by diffeos outside $K_n$. So the resulting topology is not Baire. Conclusion: there is no Baire metrizable group topology for which these are the converging sequences. | |
| Nov 13, 2023 at 8:23 | comment | added | HUO | Thanks, I added this assumption. | |
| Nov 13, 2023 at 8:20 | history | edited | HUO | CC BY-SA 4.0 |
In light of the comment.
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| Nov 12, 2023 at 23:35 | comment | added | David Roberts♦ | If $M$ is $\sigma$-compact (equiv. to second countable here) or even merely paracompact, then you might be able to get a stronger result. Not massively useful, but maybe one of these is assumed in your definition of finite-dimensional manifold. | |
| Nov 12, 2023 at 23:20 | history | asked | HUO | CC BY-SA 4.0 |