Timeline for Existing proofs of Rokhlin's theorem for PL manifolds
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 27, 2012 at 16:11 | vote | accept | Vitali Kapovitch | ||
| Feb 10, 2012 at 14:41 | comment | added | Vitali Kapovitch | @Andy: yes, your observation does prove the existence part. It does not prove uniqueness which I think is much harder to prove. but as you say, the uniqueness part is not needed here. | |
| Feb 10, 2012 at 5:47 | comment | added | Andy Putman | @Vitali Kapovitch : Isn't the handlebody observation I described essentially equivalent to PL=Diff in dimension 4 (the existence part, not the uniqueness part; but all you need is existence). | |
| Feb 10, 2012 at 5:27 | comment | added | Vitali Kapovitch | @Andy: oh, I definitely knew that the result was correct. what I wanted was a clear proof of the theorem for PL manifolds if possible avoiding unnecessary machinery such as the fact that PL=Diff in dimension 4 which I think is a much harder result than Rokhlin's theorem itself. However, it seems that all one needs is that TOP=PL=DIFF in dimension 3 which I'm ok with using. In any case I like the proof your describe very much as it really clarifies things. | |
| Feb 10, 2012 at 5:02 | comment | added | Andy Putman | @Vitali : Yes, that's one way to do it. I wasn't sure what you wanted in your question -- I had assumed that you wanted a PL proof for aesthetic reasons or something. But if all you care about is the correctness of the result, then this is all you need. | |
| Feb 10, 2012 at 4:52 | comment | added | Vitali Kapovitch | @Andy Putman: that's a very nice observation! I didn't realize that. Of course this being the case my original question becomes somewhat moot: once one proves that a PL 4-manifold has a handlebody decomposition (which I think is obvious) it is then smoothable and hence the smooth Rokhlin's theorem applies. | |
| Feb 10, 2012 at 4:20 | comment | added | Andy Putman | @Vitali Kapovich : Not always. In fact, if a 4-manifold has a handle decomposition, then it is smoothable. The point is that the attaching maps are homeomorphisms of $3$-manifolds onto their images, and such maps are always isotopic to smooth maps. There's a brief discussion of this in Chapter 9.2 of Freedman and Quinn's book. | |
| Feb 10, 2012 at 4:00 | comment | added | Vitali Kapovitch | Thanks, I'll certainly take a look at that paper. I really want to see a clean PL proof that makes it clear where exactly the PL structure is used and why the proof fails in the TOP category. From what you are saying it semms that in the approach you describe this happens on the handlebody decomposition step. It's still far from clear to me though because I think topological handlebody decomposition always exists by Freedman, isn't this right? | |
| Feb 10, 2012 at 1:18 | history | answered | Andy Putman | CC BY-SA 3.0 |