Timeline for Can all n-manifolds be obtained by gluing finitely many blocks?
Current License: CC BY-SA 2.5
5 events
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| Oct 18, 2010 at 16:40 | history | edited | HJRW | CC BY-SA 2.5 |
Acknowledged mistake, added something hopefully useful
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| Oct 18, 2010 at 15:56 | comment | added | Greg Kuperberg | He's not allowing gluing with ridges, only gluing along boundary. | |
| Oct 18, 2010 at 15:32 | comment | added | Bruno Martelli | It might be worth saying that if $S= \{M_1,\ldots , M_k\}$ consists of manifolds with toric boundaries, then any hyperbolic manifold generated by $S$ has volume not bigger than the maximum Gromov norm of the $M_i$'s. Therefore you cannot get all hyperbolic 3-manifolds with finitely many cusped ones. Things however are more complicate if we admit (as we do) higher-genus boundary. | |
| Oct 18, 2010 at 15:26 | comment | added | Bruno Martelli | You are right that $M_8$ is universal, but its coverings are not obtained by gluing copies of $M_8$. If you construct a manifold by gluing copies of $M_8$ then you never get a hyperbolic manifold, because it contains many incompressible tori. (On the other hand, all coverings of $M_8$ are hyperbolic.) | |
| Oct 18, 2010 at 15:19 | history | answered | HJRW | CC BY-SA 2.5 |