Timeline for Example of open manifold with no free integer homology non-homeomorphic to a ball
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 28, 2018 at 13:05 | comment | added | Liviu Nicolaescu | In dimension 4 there are the so called Mazur manifolds that are contractible and not diffeomorphic to the ball. en.wikipedia.org/wiki/Mazur_manifold | |
| Oct 27, 2018 at 18:55 | comment | added | mme | @Igor Thank you for the correction. I need to rely on something more precise than my weak memory! | |
| Oct 27, 2018 at 18:53 | comment | added | Igor Belegradek | @MikeMiller: it is actually Stallings. The reference is in en.wikipedia.org/wiki/Simply_connected_at_infinity. Siebenman of course also worked on this topic starting from his thesis, see citeseerx.ist.psu.edu/viewdoc/…. | |
| Oct 27, 2018 at 18:37 | comment | added | mme | Siebenmann! Not Stallings. My mistake. | |
| Oct 27, 2018 at 16:40 | comment | added | mme | In dimension 2 every simply connected manifold is homeomorphic to either the 2-sphere or the ball. | |
| Oct 27, 2018 at 16:39 | comment | added | mme | There are contractible manifolds in every dimension at least 3 whose 'fundamental group at infinity' is not trivial, hence are not homeomorphic to a ball. It is a theorem of Stallings that any contractible manifold of dimension at least 5 with trivial fundamental group at infinity is homeomorphic to the ball, and a theorem of smoothing theory (unfortunately I don't know the names to cite here) that one may even say diffeomorphic there. | |
| Oct 27, 2018 at 16:28 | comment | added | MathBug | Sure, stupid question. I was going to delete it. | |
| Oct 27, 2018 at 16:21 | comment | added | Najib Idrissi | @RGC Yes, it does kill the last homology group. For a higher dimensional example, just take the product with $\mathbb{R}^n$. | |
| Oct 27, 2018 at 16:20 | review | Close votes | |||
| Nov 1, 2018 at 3:05 | |||||
| Oct 27, 2018 at 16:19 | comment | added | MathBug | So is there a general result for higher dimensions? $dimM=4$ is still a low-dimensional case. | |
| Oct 27, 2018 at 16:14 | comment | added | MathBug | You are right, thank you very much. | |
| Oct 27, 2018 at 16:08 | comment | added | Anubhav Mukherjee | They are simply connected, so definitely oriented. | |
| Oct 27, 2018 at 16:04 | comment | added | MathBug | Does that kill the last homology group, Najib? | |
| Oct 27, 2018 at 16:01 | comment | added | Najib Idrissi | Or simply remove a point from the Poincaré sphere, no? The result has nontrivial $\pi_1$ so it's certainly not homeomorphic to an open ball. | |
| Oct 27, 2018 at 16:01 | comment | added | MathBug | Are there examples that are oriented? | |
| Oct 27, 2018 at 15:37 | comment | added | Anubhav Mukherjee | Freedman proved that every homology sphere bounfs a contractible 4 topological manifolds. Consider such a smooth one and take it's boundary out. Now the resultant manifold is not open ball. | |
| Oct 27, 2018 at 15:20 | review | First posts | |||
| Oct 27, 2018 at 16:21 | |||||
| Oct 27, 2018 at 15:17 | history | asked | MathBug | CC BY-SA 4.0 |