Timeline for Can the loops in the definition of the fundamental group be considered injective?
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
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| Nov 10, 2021 at 22:30 | vote | accept | Arshak Aivazian | ||
| Nov 10, 2021 at 21:58 | answer | added | Jeremy Brazas | timeline score: 4 | |
| Nov 10, 2021 at 20:37 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Nov 10, 2021 at 20:36 | comment | added | Arshak Aivazian | @NoahSchweber Indeed, this is a typo, thanks. | |
| Nov 10, 2021 at 17:32 | comment | added | Noah Schweber | @EschatumVerus I think you misunderstood my comment. In the question, you define the notion of essentially wide ("We say that $D\subset C$ is essentially wide if [...]") but you then use the phrase homotopically wide later on (e.g. "Is there a $D$ homotopically wide subclass [...]"), and indeed the word "essentially" never appears again. Do you really want to use both phrases? (And meanwhile you never use "substantially wide" except in your comment.) | |
| Nov 10, 2021 at 13:06 | comment | added | Will Sawin | What about passing from $X$ to $X \times \mathbb R^3$ or something? | |
| Nov 10, 2021 at 6:43 | answer | added | Achim Krause | timeline score: 1 | |
| Nov 10, 2021 at 5:54 | comment | added | Arshak Aivazian | @NoahSchweber By using "substantially wide" I refer to the concept of a substantially wide subcategory. If you think it was worth adding "homotopically" to make it clear what morphisms we are talking about, then I will add it now. | |
| Nov 10, 2021 at 5:48 | comment | added | Noah Schweber | Should "essentially wide" be "homotopically wide"? | |
| Nov 10, 2021 at 5:23 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Nov 10, 2021 at 5:02 | comment | added | Arshak Aivazian | @WillSawin Indeed, thanks, but this is an insignificant problem in the sense of "let us pass to a homotopically equivalent space from this class of spaces". Now I will make changes to the question. | |
| Nov 10, 2021 at 4:18 | comment | added | Will Sawin | Try a circle with a line sticking out at one point, with the base point on that line. | |
| Nov 10, 2021 at 3:36 | comment | added | Arshak Aivazian | @RyanBudney Thank you | |
| Nov 10, 2021 at 3:34 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Nov 10, 2021 at 3:32 | comment | added | Arshak Aivazian | Oh, the next obvious counter-example is a bunch of two circles. In fact, I meant that any words (i.e., elements of a free group) from injective loops are allowed. Now the question is formulated correctly. | |
| Nov 10, 2021 at 3:30 | comment | added | Ryan Budney | It's called the weak Whitney embedding theorem. Wikipedia has a page with references. | |
| Nov 10, 2021 at 3:28 | comment | added | Arshak Aivazian | @skupers I'm interested arbitrary subsets of $\mathbb {R}^3$, not necessarily manifolds. Nevertheless, thanks, can you tell me where I can read about the fact you mentioned? | |
| Nov 10, 2021 at 3:23 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Nov 10, 2021 at 3:19 | comment | added | skupers | On the other hand, if $X$ is a manifold dimension at least $3$ you can make any loop embedded by general position. You can similarly make homotopies go through embeddings if the dimension is at least $4$. | |
| Nov 10, 2021 at 3:19 | comment | added | Arshak Aivazian | @SimonHenry Really! It is necessary to extend the class of loops: this is the closure of the class of injective loops with respect to the exponentiation operation. I'll edit the question now, thanks. | |
| Nov 10, 2021 at 3:12 | comment | added | Simon Henry | Isn't the answer obviously no with $X= S^1$ ? there are few (Homotopy class of) injective map $S^1 \to S^1$ | |
| Nov 10, 2021 at 2:58 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |