Timeline for Low dimensional topological manifolds
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2016 at 11:48 | comment | added | Tom | It is provable in ZFC that there are $2^{\aleph_1}$ pairwise non-homeomorphic non-metrizable Hausdorff-surfaces: Just take different smooth structures on the Open Long Ray and look at the tangent bundles with respect to the different smooth strucutres. Nyikos showed in [Various Smoothings of the Long Line] that you can get 2^aleph1 many non-homeomorphic 2-manifolds just by this very special construction. | |
| Mar 28, 2014 at 15:34 | comment | added | Mathieu Baillif | And let me add that apparently the classification of $1$-manifolds was first worked out by Hellmuth Kneser in "Sur les variétés connexes de dimension 1, Bull. Soc. Math. Belg. 10 (1958)" (though it was certainly known before, but it seems nobody had written it down). | |
| Mar 28, 2014 at 15:30 | comment | added | Mathieu Baillif | without copy of $\omega_1$, and PFA implies that a longpipe possesses a copy of $\omega_1$ (due to Balogh, I think). And yes, there are $2^{\aleph_1}$ non pairwise homeomorphic surfaces, as shown by Nyikos in the same article. | |
| Mar 28, 2014 at 15:26 | comment | added | Mathieu Baillif | Actually, there are long pipes that do not contain a copy of the long line, for instance take a tangent bundle of the longline, remove the $0$ section and take one of the two components. This does not contain a copy of the longline but is not $\omega$-bounded. You can then take the quotient under the $\mathbb{Z}$-action $n\mapsto 2^n\cdot x$ on each fiber, this gives a longpipe without embedded longline. What is ZFC-independant is whether there is a copy of $\omega_1$ in each such longpipe. The paper by Nyikos in the Handbook contains a construction under $\diamondsuit$ of a longpipe (cont...) | |
| Mar 28, 2014 at 8:29 | history | edited | Benoît Kloeckner | CC BY-SA 3.0 |
added 12 characters in body
|
| Mar 27, 2014 at 20:44 | history | answered | Benoît Kloeckner | CC BY-SA 3.0 |