Timeline for Griffiths and Harris reference
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Oct 14, 2015 at 14:40 | history | suggested | David Steinberg | CC BY-SA 3.0 |
edit for gender-neutral language
|
| Oct 14, 2015 at 14:28 | review | Suggested edits | |||
| S Oct 14, 2015 at 14:40 | |||||
| Oct 14, 2015 at 8:10 | comment | added | Thomas Rot | Just a small remark how the orientability enters, since this confused me. The Morse complex depends on a choice of orientations of the unstable manifolds. If the manifold is oriented, this choice also orients the stable manifolds canonically. Hence the negative function comes with a natural choice of orientation of the unstable manifolds (which are the stable ones of the original function). This is of course folklore, and I wrote some more details in chapter four of my phd thesis (also for manifolds with boundary). | |
| Oct 14, 2015 at 4:18 | history | edited | roy smith | CC BY-SA 3.0 |
added 1 character in body
|
| Jan 15, 2011 at 23:06 | history | edited | roy smith | CC BY-SA 2.5 |
added 100 characters in body
|
| Dec 12, 2010 at 19:52 | comment | added | roy smith | According to Clint McCrory, the expert I asked, the Goresky Macpherson paper employs a generalized version of the classical simplicial "dual cell" argument, which does have classical PD as a corollary. But a classical such exposition, if one can be found, may be more accessible. | |
| Dec 12, 2010 at 17:20 | comment | added | roy smith | Vamsi, you raise a good question - just what does "Poincare duality" say? The usual treatments are via pairings on functionals, such as wedge products on forms and de Rham cohomology, or cup and cap products in cohomology and homology. The intuitively desirable geometric approach via intersection pairings on homology, as attempted in GH, is apparently more difficult to carry out, [and ignores torsion]. My search has found no satisfactory printed treatment so far along the lines of what you ask, but I have not seen the paper of Goresky and Macpherson mentioned in another answer here. | |
| Dec 11, 2010 at 23:06 | comment | added | Dave Anderson | @David, there's more to check, but the Morse argument does essentially prove the actual duality statement: use the fact that the "flow-up" and "flow-down" manifolds intersect transversally in the critical point. | |
| Dec 11, 2010 at 21:17 | comment | added | Sheikraisinrollbank | Where does the Morse theoretic argument really use that $M$ is a manifold? Just in the definition of Morse function? Natural follow up: will the same argument work for stratified spaces/intersection homology? | |
| Dec 11, 2010 at 19:13 | comment | added | Vamsi | The answer(s) and comments are quite enlightening (and the morse theory proof is very beautiful). However, my question is slightly different. GH first define the intersection number of two smooth transverse cycles, then they "prove" that this is well-defined at the level of homology and then attempt to prove Poincare duality from this. My question is, is there a place where PD is done this way? | |
| Dec 11, 2010 at 18:10 | comment | added | David E Speyer | It seems to me that this argument proves that $\dim H^k(X) = \dim H^{n-k}(X)$, but it doesn't prove that $H^k(X) \times H^{n-k}(X) \to H^n(X)$ is a perfect pairing. Or am I missing something? | |
| Dec 11, 2010 at 16:51 | history | edited | roy smith | CC BY-SA 2.5 |
deleted 218 characters in body
|
| Dec 11, 2010 at 16:34 | comment | added | Donu Arapura | I remember learning such a proof in Morgan's topology class in the 80's. The details are quite fuzzy right now, but I don't recall a gradient flow, it was pure handlebody theory. | |
| Dec 11, 2010 at 15:51 | comment | added | Deane Yang | Roy and Jim, thanks for the clarifications. So there is actually quite a bit more than just flipping the map. But it's still a beautiful proof, even if it is, as Andy points out, morally the same as the triangulation proof. | |
| Dec 11, 2010 at 7:42 | comment | added | Jim Bryan | I think the missing "details" in the Morse function proof of PD is actually constructing (co)homology from a Morse function. To do this you have to construct a differential out of the gradient flow of the Morse function. This can be done rigorously (the wikipedia page on Morse homology has some references), but it is considerably more difficult than the standard treatment of Morse theory such as what appears in Milnor's book. Once you have Morse homology though, then yes, Poincare duality is simply change the sign of the Morse function. | |
| Dec 11, 2010 at 7:25 | comment | added | roy smith | nice remark Andy! | |
| Dec 11, 2010 at 6:29 | comment | added | Andy Putman | I think the "Morse function proof" is morally the same as the "dual triangulation proof". Indeed, a triangulation can be thickened up to a handle decomposition, thus giving you a Morse function which when you "turn it upside down" gives you the handle decomposition of the dual triangulation. | |
| Dec 11, 2010 at 5:17 | comment | added | roy smith | well perhaps my phrase "we are done" is exaggerated. presumably there are some "details".... | |
| Dec 11, 2010 at 4:45 | comment | added | Deane Yang | My gosh. Is the proof of Poincare duality using a Morse function really that easy and obvious? Why is this not taught to absolutely everyone? It sounds like Milnor himself does not mention it in his notes. How could that be? This is by far the most stunning thing I've learned on MathOverflow. | |
| Dec 11, 2010 at 4:29 | history | answered | roy smith | CC BY-SA 2.5 |