Timeline for Are non-PL manifolds CW-complexes?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 9, 2016 at 22:00 | comment | added | Danu | @IgorBelegradek Thank you for the clarifications. | |
| Aug 9, 2016 at 21:35 | comment | added | Igor Belegradek | @Danu: Kirby-Siebenmann's book p.107: every closed topological manifold of dimension $\ge 6$ is homeomorphic to a CW complex. In the comment above I was of course talking about higher dimensions. The statement that every manifold (compact or not) is homotopy equivalent to a CW complex is much easier. For compact manifolds this is proved e.g. in Hatcher's textbook (in appendix on CW complexes). | |
| Aug 9, 2016 at 20:39 | comment | added | Danu | @IgorBelegradek That doesn't sound right: The paper by Kirby-Siebenmann claims that they have the homotopy type of a CW complex, not that they are homeomorphic. Furthermore, the answers to this MSE post seem to provide references that the case $n=4$ is still open. I don't know much (if anything) about these topics, but could you provide a reference to back up that claim? | |
| Aug 27, 2010 at 13:15 | comment | added | Igor Belegradek | @Ryan, the open problem is not whether any compact manifold is homeomorphic to a CW complex (this was proved by Kirby-Siebenmann). The open problem is whether it has a (non-combinatorial) triangulation. @grad student, whatever is known in the noncompact case must be Kirby-Siebenmann's book. | |
| Aug 27, 2010 at 4:30 | comment | added | Ryan Budney | @Dev, yes, that sounds right. As algori mentions that's apparently an open problem. As far as I know that problem hasn't attracted a whole lot of attention. | |
| Aug 27, 2010 at 4:27 | comment | added | A grad student | @Ryan : Yes, I think that is what Milnor proved (it's also been a long time since I looked at it). | |
| Aug 27, 2010 at 4:23 | comment | added | Ryan Budney | It's been a while since I've looked at that Milnor paper -- I suspect maybe he's arguing that manifolds have the homotopy-type of countable CWs, while Kirby-Siebenmann deal with compact manifolds and finite CWs. ? | |
| Aug 27, 2010 at 4:22 | comment | added | Dev Sinha | But I thought the question was whether each has the "homeomorphism type" of a CW complex. | |
| Aug 27, 2010 at 4:08 | comment | added | A grad student | I think the fact that they have the homotopy type of a CW complex is due to Milnor (it is in his paper about spaces homotopy equivalent to CW complexes). Do Kirby-Siebenmann just prove this, or do they prove that all compact manifolds are homeomorphic to CW complexes? Also, how about the noncompact case? | |
| Aug 27, 2010 at 4:04 | history | answered | Ryan Budney | CC BY-SA 2.5 |