Timeline for Are there any tests for knowing whether a topological space admits a CW structure?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 4 at 9:38 | vote | accept | Tyrannosaurus | ||
| Dec 3 at 11:39 | comment | added | Moishe Kohan | See also my answer here. | |
| Dec 3 at 5:28 | comment | added | Moishe Kohan | As for recognizing topological manifolds, there is a complete list of conditions in dimension $\ge 5$: Metrizable, locally contractible, finite dimensional, homology manifold, DDP and vanishing local index (Cannon, Edwards, Quinn,...) | |
| Dec 2 at 16:50 | history | became hot network question | |||
| Dec 2 at 15:32 | comment | added | Tyrannosaurus | @DenisT By manifold, I meant the ones which are metrisable and by admitting CW structure, I meant homeomorphism. | |
| Dec 2 at 15:13 | comment | added | Denis T | Also, "admitting CW structure" can be understood at least in two different ways: being homeomorphic to a CW, or being homotopy equivalent. (It is not clear whether you're interested in the setting of general topology or algebraic topology.) | |
| Dec 2 at 14:54 | comment | added | Denis T | Can you provide precise definition of a manifold you're using? For example, if you do not require metrizability, then you have an example by Calabi-Rosenlicht (which is separable and Hausdorff), which can be informally described as a sphere with continuum-many punctures along the big circle. It is weakly equivalent to a CW complex, but not homotopy equivalent. So, metrisability is necessary (and also sufficient, by Milnor). | |
| Dec 2 at 10:06 | answer | added | Francesco Polizzi | timeline score: 10 | |
| Dec 2 at 9:27 | comment | added | HJRW | I believe the question of whether or not every finitely presented Poincaré Duality group is the fundamental group of a closed aspherical manifold is open in all dimensions >2. So the "obstructions that can arise that stop a CW complex being a manifold" are not fully understood (though, as you say, Poincaré duality is certainly a necessary condition). | |
| S Dec 2 at 8:49 | review | First questions | |||
| Dec 2 at 9:37 | |||||
| S Dec 2 at 8:49 | history | asked | Tyrannosaurus | CC BY-SA 4.0 |