Timeline for Topological Classification of Four-Manifolds
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 23, 2018 at 6:38 | comment | added | QGravity | @FanZheng Thank you for the comment. In general relativity, we should deal with the space of Lorentzian four-metrics on a fixed 4-manifolds $X$, and to have casual structure, we demand global hyperbolicity condition on $X$. So $X=\mathbb{R}\times W$ for a Riemannian 3-manifold $W$. Does this mean there is some sort of equivalence between the space of Lorentzian 4-metrics on $X$ and the space of Riemannian 3-metrics on $W$? | |
| Jul 22, 2018 at 4:09 | comment | added | Fan Zheng | @QGravity If a 4-manifold is globally hyperbolic, then it is diffeomorphic to $M\times\mathbb R$ where $M$ is a 3-manifold (whose classification is more tractible), see en.wikipedia.org/wiki/Globally_hyperbolic_manifold. | |
| Feb 25, 2018 at 0:48 | comment | added | QGravity | I guess the topological constraint(s) come from global hyperbolicity condition. | |
| Feb 21, 2018 at 22:11 | comment | added | Paul Siegel | In physics applications you have a lot more structure, like a Minkowski metric with some conditions on its curvature. This extra structure can significantly restrict the topology of the underlying manifold, so it is entirely possible that you can classify all 4-manifolds which could plausibly appear in GR. (For instance, a topological 4-manifold need not have a smooth structure let alone a smooth metric.) So you are probably poking around in the wrong part of mathematics given your motivation. | |
| Feb 21, 2018 at 21:56 | comment | added | Paul Siegel | @QGravity en.wikipedia.org/wiki/… | |
| Feb 21, 2018 at 21:08 | comment | added | QGravity | In physics, the people are more interested to consider compact four-manifolds with boundaries (which might have some sort of "topological singularity" representing singular behavior in space-time like black hole singularity or big bang). I was very curious about the development in the topological classification of such four-manifolds. | |
| Feb 21, 2018 at 21:01 | comment | added | QGravity | So, isn't it possible to classify finitely-presented groups just like finitely-generated Fuchsian groups? | |
| Feb 21, 2018 at 20:54 | vote | accept | QGravity | ||
| Feb 21, 2018 at 20:54 | |||||
| Feb 21, 2018 at 10:13 | comment | added | Daniele Zuddas | Regarding your last sentence, for all manifolds of dimension $\geq 3$, $\pi_1(M) = \pi_1(M- pt)$. In addition, open 4-manifolds can have an arbitrary countable fundamental group. | |
| Feb 21, 2018 at 8:23 | history | answered | Paul Siegel | CC BY-SA 3.0 |