Timeline for Compactification of open manifolds in the form of a manifold( with zero Euler characteristic)
Current License: CC BY-SA 3.0
25 events
| when toggle format | what | by | license | comment | |
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| Mar 23, 2019 at 17:17 | vote | accept | Ali Taghavi | ||
| S Nov 20, 2017 at 11:12 | history | bounty ended | CommunityBot | ||
| S Nov 20, 2017 at 11:12 | history | notice removed | CommunityBot | ||
| Nov 12, 2017 at 11:22 | comment | added | Nick L | No it is trivial in homology but non-trivial in the fundamenetal group. This is discussed here math.stackexchange.com/questions/1031069/…. | |
| S Nov 12, 2017 at 9:39 | history | bounty started | Ali Taghavi | ||
| S Nov 12, 2017 at 9:39 | history | notice added | Ali Taghavi | Draw attention | |
| Nov 12, 2017 at 9:36 | comment | added | Ali Taghavi | @NickL But I think that the curve you mentioned is a trivial loop. Am I correct? | |
| Nov 5, 2017 at 17:27 | comment | added | Nick L | Probably it would have been more accurate to say "the class of this curve doesn't die in the embedding" (This argument wouldn't rely on the image being dense). | |
| Nov 5, 2017 at 17:22 | comment | added | Nick L | If one connect sums two tori to get a genus two surface, then this separating curve is the curve that corresponds to the boundary of the two disks that were cut out to make the connect sum. | |
| Nov 5, 2017 at 17:13 | comment | added | Ali Taghavi | @NickL But the comment of Prof. Goodwillie says there is no an open set of the torus homeomorphic to the $M_2 \setminus \{pt\}$. He does not consider compactification. BTW waht is that separating non trivial closed curve on $M_2$. | |
| Nov 5, 2017 at 16:52 | comment | added | Nick L | Perhaps use the fact that the genus two surface contains a separating curve which is non-trivial in $\pi_{1}$, You just have to prove that the class of this curve doesn't die in the compactification. | |
| Nov 5, 2017 at 11:53 | history | edited | Ali Taghavi |
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| Nov 5, 2017 at 11:51 | comment | added | Ali Taghavi | @TomGoodwillie what is the reason of this non embedding property?Are you using relative homology? However this non embedding is obvious intutively but i can not find a proof for that.Thanks for your attention to my question. | |
| Nov 4, 2017 at 12:38 | comment | added | Tom Goodwillie | No. The complement of a point in an oriented surface of genus 2 cannot be embedded in a torus or a Klein bottle. | |
| Nov 3, 2017 at 21:57 | answer | added | Nick L | timeline score: 8 | |
| Nov 3, 2017 at 21:23 | history | edited | Ali Taghavi |
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| Nov 3, 2017 at 21:09 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Nov 3, 2017 at 21:08 | review | Close votes | |||
| Nov 4, 2017 at 13:22 | |||||
| Nov 3, 2017 at 20:59 | comment | added | Ali Taghavi | @MichaelAlbanese Thank you for sharing the link. I revise the question with emphasize on on "zero Euler characteristic. | |
| Nov 3, 2017 at 20:49 | comment | added | Michael Albanese | Possible duplicate of Compactification of a manifold | |
| Nov 3, 2017 at 20:35 | comment | added | Nick L | I guess you want to assume that $M$ has finite topological type (for example it has CW complex structure with finite number of cells). Otherwise it can never embed in a compact manifold (consider surface with infinite genus for example). | |
| Nov 3, 2017 at 20:34 | comment | added | Neal | @MichaelAlbanese I was thinking too rigidly. Thank you. | |
| Nov 3, 2017 at 20:32 | comment | added | Michael Albanese | @Neal: Once you remove the pinch point, what's left is homeomorphic to a cylinder which embeds in a torus. | |
| Nov 3, 2017 at 20:30 | comment | added | Neal | Pinch a torus and remove the pinch point. The result is an open connected manifold. How would you compactify this so that the result is a manifold? | |
| Nov 3, 2017 at 20:25 | history | asked | Ali Taghavi | CC BY-SA 3.0 |