Timeline for Homotopy groups of spaces of embeddings
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2021 at 15:16 | history | edited | archipelago |
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| Mar 26, 2021 at 14:08 | answer | added | skupers | timeline score: 10 | |
| May 28, 2014 at 3:58 | answer | added | Tom Goodwillie | timeline score: 13 | |
| May 27, 2014 at 22:58 | history | edited | John Klein |
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| May 27, 2014 at 22:28 | answer | added | John Klein | timeline score: 7 | |
| May 27, 2014 at 14:32 | comment | added | Igor Belegradek | @MarkGrant: Yes, I know there is a rich literature when $N=\mathbb R^n$, yet I do not know what makes this case special (other than the fact that people find knots interesting). For example, is Dax's result known to fail below the metastable range? | |
| May 27, 2014 at 14:27 | comment | added | Igor Belegradek | @RicardoAndrade: my base point is a homotopy equivalence (as I stressed above), so its connectivity is not an issue. | |
| May 27, 2014 at 14:20 | comment | added | Ricardo Andrade | Dear @Igor Belegradek: If you meant the result stated at the top of the page numbered 305 in Dax's article, then I think you have left out one necessary condition: the connectivity of the map $M\to N$ which you take as the basepoint must be at least $2\dim M-\dim N +k+1$. Since you are taking arbitrary maps, you must then impose the condition $2\dim M-\dim N +k+1\leq 0$, i.e. $k\leq\dim N-2\dim M -1$, which is exactly the estimate I wrote in my previous comment. By the way, you will find precisely this connectivity estimate on the first page of the introduction of Dax's article. | |
| May 27, 2014 at 13:26 | comment | added | Mark Grant | There is some literature on the rational homotopy of spaces of long knots (embeddings $\mathbb{R}^m$ to $\mathbb{R}^n$ fixed outside a compact set $C\subset \mathbb{R}^m$). In that case $M$ is not closed, of course, but at least $M\to N$ is a homotopy equivalence. I gather people are trying to extend these results to $\operatorname{Emb}(M,N)$. You might glean some information on the current (4 years ago) state-of-the-art by looking at the list of open problems on Ismar Volic's web page, palmer.wellesley.edu/~ivolic/pdf/Papers/… | |
| May 27, 2014 at 11:43 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| May 27, 2014 at 11:43 | comment | added | Igor Belegradek | @RicardoAndrade: I added a link to Dax's paper. The result is mentioned on the second page of the introduction. There were earlier results, but Dax's is the best available, I think. | |
| May 27, 2014 at 4:24 | comment | added | Ricardo Andrade | I think the map $\operatorname{Emb}(M,N) \to \operatorname{Map}(M,N)$ is actually just $(\dim N - 2 \dim M - 1)$-connected, which should follow from some parametrized version of approximation of continuous maps by embeddings. This most likely predates Dax. By the way, I think the first bound you give is for something else (which does appear to be due to Dax) related to Haefliger's refined theory: see theorem 1.2.1 and the following remark in the survey you link to. | |
| May 26, 2014 at 2:45 | history | asked | Igor Belegradek | CC BY-SA 3.0 |