Several months ago a paper was posted at http://arxiv.org/abs/1001.4164 called "Another way of answering Henri Poincaré's fundamental question." The author gave a talk on it today at my institution. If it's correct, it is a major breakthrough in terms of proof length (~10 pages). However, it is very outside my specialty. There's apparently been very little feedback, but the author is ok with public discussion. Therefore, can anyone say whether they have read the paper? Whether it is correct, missing details, clearly flawed, or what have you?
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36$\begingroup$ I'm not hugely encouraged by the final sentence of the introduction to the paper: "This essay is devoted to the meditation of God’s Word among the inhabitants of the earth." $\endgroup$– gowersCommented May 6, 2010 at 17:38
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1$\begingroup$ Did the author indicate what feedback he's gotten from experts who have read his paper? $\endgroup$– Deane YangCommented May 6, 2010 at 17:45
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21$\begingroup$ While I would bet a large sum of money that the paper has a fatal error, I'm not comfortable with criticizing the phrase "This essay is devoted to the meditation of God’s Word among the inhabitants of the earth". While I've certainly seen crackpots make dedications like this, there are plenty of serious papers with religious dedications. The author, btw, is definitely a real mathematician (he is a professor at Universitat Bern, and according to mathscinet he has published 34 papers since the late '60's, mostly in combinatorial topology and convex geometry). $\endgroup$– Andy PutmanCommented May 6, 2010 at 21:15
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2$\begingroup$ Actually, he appears to be retired. A colleague of mine points out that the author is quite well-known for his elegant solution to the shelling problem in the theory of polytopes. So he cannot be dismissed easily. $\endgroup$– Deane YangCommented May 6, 2010 at 22:29
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2$\begingroup$ I agree that the God sentence is not evidence enough on its own. But there is another important piece of evidence: the paper is short and has been in the public domain since January. If it were correct, it would surely have been hailed as a spectacular twist in the story of the Poincaré conjecture. Given that it hasn't, I am predisposed not to believe the paper, and against that background the God sentence increases my scepticism. $\endgroup$– gowersCommented May 7, 2010 at 13:38
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2 Answers
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I had a quick look. Although I haven't found a specific error, as far as I can tell, he's not using the hypothesis of simple-connectivity anywhere in an essential way. Even though he posits this as a hypothesis in Prop. 5.6, the proof of this proposition works for any manifold with 2-sphere boundary (also, Prop. 4.4 works for any homology ball). Thus, in Step (4) of his proof of Prop. 5.8, when he refers to Prop. 5.6, this step is not making essential use of simple-connectivity.
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I also Had a quick look (maybe a little less quick), and although I very much like the other answer, which illustrates that it may be difficult to fix, I may have found a more specific error, which may be more helpfull as an answer to the question.
Firstly, I am a little confused as to what constitutes a stratification. I see two possibilities:
1) The one which is actually defined which allows the following stratification: $S_1=S_2=D^2\times [0,1]$ and they are glued along a closed disc in the interior of $D^2\times \{1\}$ of $S_1$ and the same disc in the interior of $D^2 \times \{0\}$ in $S_2$.
2) The one which I think is implied at some points: $S_i$ and $S_{i+1}$ may only be identified such that $U(S_i) \cup L(S_{i+1})$ is in fact a sub-surface in the 3-manifold.
I will describe my problems related to both definitions:
In the proof of prop 5.8 parts (2-3-4) he attaches "3-cell"s (I would write 3-disc as to avoid confusion with CW complex attachments of cells, or attach both a 2-cell and a 3-cell) $W$, and extends the stratification.
If we work under definition 2) above then this seems generally impossible because you would often also have to attach it at the top of $S_{i+1}$ to get the extra surface assumption in 2).
If we work under definition 1) above then this doesn't even make the new $F_{i+1}$ a surface in the simple example described above.