Timeline for Permuting $n$ points in a $2$-manifold
Current License: CC BY-SA 4.0
19 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Jan 24, 2020 at 14:28 | vote | accept | Meths | ||
| Jan 24, 2020 at 1:37 | answer | added | Ty Ghaswala | timeline score: 3 | |
| Jan 23, 2020 at 15:20 | comment | added | Meths | @MarkSapir That's unfortunate - the description I gave was intended to match that of the Definition given in Section 2.2 of the linked paper. I believe $\text{Mot}_{n}(M)$ is then $\pi_{1}(C_{n}(M)/S_{n})$ (which the above discussion helped me understand). According to the paper, this is indeed $B_{n}(M)$. I guess that just leaves question 2(ii). | |
| Jan 23, 2020 at 15:10 | comment | added | user6976 | After the clarification, the question has become less clear. | |
| Jan 23, 2020 at 14:42 | comment | added | Meths | Ah, I see. That makes sense if $T$ is an unordered set (which is implicit in your description). So these $n$ curves in $M$ just look like one loop in $C_{n}(M)$. If $T$ was ordered, then you'd only get a loop for the trivial permutation? I guess there are some subtleties about $M$ being path-connected meaning $C_{n}(M)$ being path-connected too (for $T$ unordered), allowing us to forget about the basepoint. | |
| Jan 23, 2020 at 14:07 | history | edited | Meths | CC BY-SA 4.0 |
Clarification 2
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| Jan 23, 2020 at 14:03 | comment | added | John Klein | Isn't your description just an equivalence class of path $\gamma: [0,1] \to C_n(M)$ such that $\gamma(0) = T = \gamma(1)$, where $C_n(M)$ is the space of subsets of cardinality $n = |T|$? | |
| Jan 23, 2020 at 14:00 | comment | added | John Klein | What you wrote in your post is still not clear: you have a curve in $M$ and you also have one in $M\times [0,1]$. Please commit and make it precise. | |
| Jan 23, 2020 at 13:58 | comment | added | Meths | *I would identify $π_1(C_n(M))$ with the homotopy equivalence classes of curves that fix $\gamma_i(0)=\gamma_i(1)$ in my description. | |
| Jan 23, 2020 at 13:39 | comment | added | Meths | I've clarified what I meant in the post now. @JohnKlein I don't have a clear understanding of why $\pi_{1}(C_{n}(M))$ doesn't restrict to the trivial permutation? | |
| Jan 23, 2020 at 13:27 | history | edited | Meths | CC BY-SA 4.0 |
Clarifying "homotopy classes of paths that permute these points".
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| Jan 23, 2020 at 10:04 | comment | added | user6976 | @Lee: The OP probably meant a path in the configuration space of $n$-tuples of points. Of course he needs to make ir clear. | |
| Jan 23, 2020 at 7:40 | review | Close votes | |||
| Jan 31, 2020 at 3:05 | |||||
| Jan 23, 2020 at 3:48 | comment | added | Lee Mosher | Without further care in formulating definitions, I do not know what it might mean for a path to permute points. I presume by a "path" you mean a continuous function $f : [0,1] \to M$, in which case "permuting points" is not something a path ordinarily does. | |
| Jan 23, 2020 at 3:21 | history | rollback | user6976 |
Rollback to Revision 1
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| Jan 23, 2020 at 2:15 | comment | added | John Klein | Questions about the definition. Let $T\subset M$ be a finite subset of cardinality $n$.This gives a basepoint for the configuration space of $n$ unordered points in $M$. Call this configuration space $C_n(M)$. Then it seems to me what you are describing is $\pi_0$ of the loop space $\Omega C_n(M)$. Is that right? If so, then your group is nothing more than $\pi_1(C_n(M))$. Right? | |
| Jan 23, 2020 at 1:10 | history | edited | YCor |
edited tags
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| Jan 23, 2020 at 1:03 | history | asked | Meths | CC BY-SA 4.0 |