Timeline for Fundamental group of the space of smooth embeddings of $S^1$ into $\mathbb R^3$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 24, 2021 at 21:57 | history | edited | Ryan Budney | CC BY-SA 4.0 |
add differential-geometric sketch of proof of the non-triviality of the suggested family.
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| Aug 24, 2021 at 21:19 | history | edited | Ryan Budney | CC BY-SA 4.0 |
add a sketch of a simpler argument to answer the question.
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| Aug 24, 2021 at 20:35 | comment | added | Ryan Budney | For $Emb(S^1, \mathbb R^3)$, yes all components have non-trivial fundamental groups. The unknot component of $Emb(S^1, \mathbb R^3)$ has the homotopy-type of $SO_3$, this is a classical result of Hatcher's but also follows from the results above. In the above description you need to use the unknot component of $K_{3,1}$ which is contractible. | |
| Aug 24, 2021 at 20:33 | comment | added | YCor | In your description, where does the precise component of the unknot appear? Does it follow from the description that every component has a trivial/nontrivial fundamental group? | |
| Aug 24, 2021 at 20:24 | history | edited | Ryan Budney | CC BY-SA 4.0 |
added 32 characters in body
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| Aug 24, 2021 at 20:19 | history | edited | Ryan Budney | CC BY-SA 4.0 |
added 32 characters in body
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| Aug 24, 2021 at 20:13 | history | answered | Ryan Budney | CC BY-SA 4.0 |