Timeline for Unstable manifolds of a Morse function give a CW complex
Current License: CC BY-SA 4.0
21 events
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Apr 15, 2020 at 0:09 | comment | added | John Klein | "My paper will be published soon. I shall tell you when it is accepted. For Laudenbach's work, I don't completely understand it. So I dare not say there are some gaps in it. However, I believe my technique is better than his partially because mine came later. Again, my technique is not due to me. The one who invented this technique did a wonderful job. The invention of this technique is much more important and creative than all subsequent proofs, including mine." | |
Apr 15, 2020 at 0:08 | comment | added | John Klein | "Laudenbach's technique is different from mine. His technique was earlier. My technique involves the compactification of moduli spaces. I have to admit that this technique was not originally proposed by me. Actually, I don't know who invented this folklore technique. Though this technique had been well-known, people hadn't completely proved certain results, for example, the CW structures arising from the Morse trajectories. Some people had made partial progress before my work. I solved this problem in the full generality firstly." | |
Apr 15, 2020 at 0:08 | comment | added | John Klein | @GaelMeigniez Note that I retracted my point (1) in the original post. I am no longer claiming the paper has gaps. Here is what Lizhen Qin wrote to me in 2019 about Laudenbach's paper (see my next two remarks). | |
Apr 14, 2020 at 7:02 | comment | added | Gael Meigniez | There is no gap in Laudenbach's paper. There are authors who, in the same time, know their subject, and are serious in what they publish. | |
Nov 26, 2019 at 12:00 | history | edited | John Klein | CC BY-SA 4.0 |
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Nov 25, 2019 at 20:06 | comment | added | John Klein | @ChrisGerig correction: Qin's third paper, "On the associativity of gluing" appeared last year. I hadn't realized it. So it is just the second paper ("An application of topological equivalence to Morse theory") on the arXiv which hasn't appeared. I do not know why. I will ask him, but as I wrote, I believe politics had something to do with it. | |
Nov 25, 2019 at 19:52 | comment | added | John Klein | @ChrisGerig I will not go on record here detailing the gaps. This is really not my mathematical subculture. You can write Lizhen Qin if you wish to know. As to why Qin's second and third papers were never published, I also suggest you write him. Suffice it to say that politics were involved. | |
Nov 25, 2019 at 18:56 | comment | added | Chris Gerig | @JohnKlein What are the gaps in Laudenbach's paper? And why was Qin's arXiv paper never published? | |
Nov 24, 2019 at 20:42 | comment | added | John Klein | Here are the steps, I believe: 1) Construct any metric which is flat at the critical points (this is what Milnor does). 2) Without changing the metric near the critical points, one can infinitesimally deform it to a Morse-Smale one (this is a generic condition). | |
Nov 24, 2019 at 18:16 | comment | added | Dmitri Panov | Unfortunately, I don't seem to find any discussion of transversality of unstable manifolds in Milnor's book. I'll try to look more carefully. I mean, I understand how to construct a metric so that it is flat close to a critical point and the gradient has this simple shape, but does Milnor also explain why the unstable manifolds are transversal for his metric? Unfortunately I can't search on "transversal" in this book, since it is not a very good pdf... | |
Nov 24, 2019 at 18:02 | comment | added | John Klein | Milnor's book gives such a metric. It's a partition of unity argument. | |
Nov 24, 2019 at 17:56 | comment | added | Dmitri Panov | Thanks, I got it! How does one prove that such a metric exists? (i.e so that Morse-Smale holds + the vector field is standard) | |
Nov 24, 2019 at 17:49 | comment | added | John Klein | Yes, that is what Laudenbach wants. It's not the metric that is the problem. There is no issue finding such a metric. The issue is with compactifying the open stable manifolds. For a critical point of index $k$, there is a smooth map $\text{int}(D^k) \to M$ whose image is the entire unstable manifold. But the problem is extending this map to a continuous map of all of $D^k$. That is the non-trivial thing. | |
Nov 24, 2019 at 17:21 | comment | added | Dmitri Panov | John, thanks for both comments. So, this is Theorem 3.8 from Section 3.4 as far as I got. It is stated that this theorem holds under assumptions of Theorem 3.4, which holds under assumptions of Theorem 3.3 :) But if I got it correctly, these assumptions mean exactly the same thing as what Laudenbach wants in case the manifold is compact? I.e. Morse-Smale holds + the vector field is standard close to fixed points | |
Nov 24, 2019 at 17:03 | history | edited | John Klein | CC BY-SA 4.0 |
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Nov 24, 2019 at 16:57 | comment | added | John Klein | Correction: it was a paper by Burghelea AND Haller. I have corrected my post. See reference [8] of the paper by Qin. | |
Nov 24, 2019 at 16:56 | history | edited | John Klein | CC BY-SA 4.0 |
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Nov 24, 2019 at 16:53 | comment | added | John Klein | Qin's paper, On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds, Theorem, Section 3.4. (As far as the existence of a "Special Morse" metric is concerned, this is not hard: use a partition of unity to construct a metric that is standard metric near the critical points. This is done in Milnor's book.) | |
Nov 24, 2019 at 16:48 | comment | added | Dmitri Panov | Dear John, thank you for this answer. I browsed this paper for some short time before asking the question (but could not quite find the result). Would you be kind to point me to the result in the paper which I could cite? And what is Burghelea's paper? | |
Nov 24, 2019 at 16:47 | history | edited | John Klein | CC BY-SA 4.0 |
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Nov 24, 2019 at 16:41 | history | answered | John Klein | CC BY-SA 4.0 |