Timeline for I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?
Current License: CC BY-SA 3.0
25 events
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| Sep 18, 2011 at 18:55 | vote | accept | Nikita Kalinin | ||
| Sep 18, 2011 at 18:55 | history | bounty ended | Nikita Kalinin | ||
| Sep 18, 2011 at 17:19 | comment | added | Ryan Budney | @Nikita: It sounds like you're now asking for the relationship between the "chart definition" of submanifold and the "regular value" definition. These are equivalent, but in Frechet manifolds the statement of "regular value" is a little subtle -- see the "Nash Moser theorem" article on Wikipedia. | |
| Sep 18, 2011 at 16:39 | history | edited | Nikita Kalinin | CC BY-SA 3.0 |
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| Sep 17, 2011 at 12:55 | answer | added | Kevin Walker | timeline score: 5 | |
| Sep 16, 2011 at 13:00 | answer | added | Andrew Stacey | timeline score: 4 | |
| Sep 16, 2011 at 6:32 | comment | added | Andrew Stacey | Nikita: I'm thinking about it. I need to consider the implications of Ryan's answer since I know that if you change the conditions ever so slightly then the answer is "yes" so I need to think carefully about what Ryan says to see how that modification affects his answer. | |
| Sep 15, 2011 at 17:13 | comment | added | Nikita Kalinin | to Andrew Stacey: I think you know answer on my question... Or you know that nobody knows. | |
| Sep 15, 2011 at 16:02 | history | edited | Nikita Kalinin | CC BY-SA 3.0 |
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| Sep 15, 2011 at 15:50 | comment | added | Andrew Stacey | ... and in every other category of generalised smooth spaces! But that doesn't mean that it has a manifold structure. | |
| Sep 15, 2011 at 15:20 | history | edited | Nikita Kalinin | CC BY-SA 3.0 |
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| Sep 15, 2011 at 7:26 | comment | added | Konrad Waldorf | In the diffeological setting, the answer is YES, for trivial reasons: every subset of a diffeological space is a diffeological subspace. | |
| Sep 14, 2011 at 18:10 | comment | added | Nikita Kalinin | in appropriate topology, near the $f_0$ there are only loops which are embeddings or loops with only one selfintersection and without other singularities. | |
| Sep 14, 2011 at 17:57 | answer | added | Ryan Budney | timeline score: 4 | |
| Sep 14, 2011 at 17:18 | comment | added | Ryan Budney | @Nikita: I'm now more unclear on what you want. You object $D$ allows for maps that are not immersions. But in your edit you say "without other singularities", and I consider not being an immersion to be a type of singularity as the derivative does not have full rank. Do you want the maps in $D$ to also be immersions? Also, are you allowing $a=b$ in your definition of $D$, or not? | |
| Sep 14, 2011 at 16:45 | comment | added | Nikita Kalinin | I mean "some neighborhood of $f_0$ in $D$". | |
| Sep 14, 2011 at 12:23 | comment | added | Nikita Kalinin | to Ryan: yes. $D = \{f\in maps(S^1,M)| \exist a,b\in S^1, f(a)=f(b)\}$. $f_0\in D$ has lone single double point. Is it true that some neighboor of $f_0$ is a submanifold of $map(S^1\to M)$? I know that near the $f_0$ there is normal bundle to $D$ (in some sense) --- you can consider versal deformation, miniversal deformation has dimension $\dim M-2$ and so on. But the question is about $D$ itself. | |
| Sep 13, 2011 at 19:42 | comment | added | Ryan Budney | Could you be a little more specific with your question? My first reading of your revised question would be that you are looking at the subspace of smooth maps $S^1 \to M$ which are immersions, and which are one-to-one with the lone exception of a single double point. Is this the subspace you are interested in? | |
| Sep 13, 2011 at 17:21 | history | edited | Nikita Kalinin | CC BY-SA 3.0 |
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| Sep 13, 2011 at 17:19 | history | bounty started | Nikita Kalinin | ||
| Sep 10, 2011 at 9:54 | history | edited | Nikita Kalinin | CC BY-SA 3.0 |
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| Sep 10, 2011 at 9:42 | history | edited | Nikita Kalinin | CC BY-SA 3.0 |
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| Sep 10, 2011 at 9:36 | history | edited | Nikita Kalinin | CC BY-SA 3.0 |
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| Sep 10, 2011 at 9:10 | answer | added | Torsten Ekedahl | timeline score: 3 | |
| Sep 10, 2011 at 7:35 | history | asked | Nikita Kalinin | CC BY-SA 3.0 |