Timeline for Simple question of topological cofibration
Current License: CC BY-SA 2.5
6 events
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| Feb 27, 2010 at 1:09 | comment | added | algori | Dan -- indeed, I was a bit imprecise on that account. But all imaginable manifolds admit a triangulation (hence, a CW-structure, but not necessarily a PL-structure). And yes, I think the standard proof of the existence of triangulations for smooth manifolds can be adapted to the case with corners. | |
| Feb 27, 2010 at 0:35 | comment | added | Dan Ramras | This seems a bit optimistic, in general. Just for ordinary manifolds, it's still an open question whether every closed topological 4-manifold has a CW structure! This is mentioned in Hatcher's book, after Corollary A.12 in the appendix. If your manifolds are smooth, then you're in better shape: all smooth manifolds (even C^1 manifolds) are triangulable. One proof is due to Cairns. Does anyone know the status of these questions for manifolds with corners? | |
| Feb 22, 2010 at 18:34 | vote | accept | mpdude | ||
| Feb 20, 2010 at 1:04 | comment | added | algori | mpdude -- if you have a submanifold (with corners) of a manifold (with corners), one can equip both with CW structures so that the submanifold becomes a subcomplex, in which case the inclusion is a cofibration. This happens whenever you have a "reasonable" space "of geometric origin" sitting inside another "reasonable" space "of genoetric origin" as a closed subspace. | |
| Feb 20, 2010 at 0:09 | comment | added | mpdude | I see, so it just boils down to the fact that I can pick some CW structure on a manifold with corners. Do I need to show that the map $X \to Y$ takes dimension $d$ faces to dimension $d$ faces? | |
| Feb 17, 2010 at 3:43 | history | answered | algori | CC BY-SA 2.5 |