Timeline for Compactification of a manifold
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2022 at 10:55 | comment | added | Aleksandar Milivojević | For future reference, this is in Siebenmann's thesis "The obstruction to finding a boundary for an open manifold of dimension greater than five", available e.g. here math.uchicago.edu/~shmuel/tom-readings/Siebenmann%20thesis.pdf | |
| Jan 19, 2018 at 0:18 | comment | added | Hadrian Quan | @BenoîtKloeckner, would you be willing to provide more information on this paper (or papers?) of Siebenmann? I'm quite interested, and having trouble finding this result. | |
| Aug 5, 2010 at 18:03 | comment | added | Benoît Kloeckner | On a slightly different setting, there is a characterization of manifolds that are the interior of a manifold with boundary; This was investigated by Larry Siebenmann and browsing its web page or MathSciNet references should help. | |
| Aug 5, 2010 at 11:39 | answer | added | damiano | timeline score: 19 | |
| Aug 5, 2010 at 10:14 | comment | added | rpotrie | You should post that as an answer! | |
| Aug 5, 2010 at 10:09 | comment | added | rpotrie | Maybe the question can be changed to: Is it possible to embed any manifold in a compact manifold in order that the image is open and dense? This seems quite "minimal" to me. | |
| Aug 5, 2010 at 10:01 | comment | added | damiano | If you attach finitely many cells to a CW complex with finite dimensional homology groups, then the resulting CW complex still has finite dimensional homology groups. Thus, if you start with a manifold with a non-finitely generated homology group, you cannot "complete" it with finitely many cells. Thus, the complement of a closed set of points in the plane with infinitely many components will do. For instance, you can remove the Cantor set in the $x$-axis from the plane. | |
| Aug 5, 2010 at 9:57 | history | edited | Charles Matthews | CC BY-SA 2.5 |
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| Aug 5, 2010 at 9:56 | answer | added | Charles Matthews | timeline score: 4 | |
| Aug 5, 2010 at 9:54 | comment | added | rpotrie | Just a remark, it is interesting to look at $\mathbb{T}^2$ times $(0,1)$ which has two ``boundary components'', but apparently the "minimal" way to embed it in a compact manifold is to include it in $\mathbb{T}^2 \times S^1$ (the other possible compactifications are "singular"). How can we define minimal though? | |
| Aug 5, 2010 at 9:41 | history | asked | Italo | CC BY-SA 2.5 |