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for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf

Because I am not able to understand the whole paper, I want to ask if someone read this paper and can someone say, if the proof is right or not.

Edit January 12, 2011: A new version is online: http://arxiv.org/pdf/math/0505634v4.

Edit October 31, 2016: Michael Atiyah has posted a paper on the arXiv claiming to prove that the complex 6-sphere does not exist. https://arxiv.org/abs/1610.09366

Tom Church
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Florian Modler
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    I'm unable to understand the paper either, but I must say the phrase "in what follows we hope to prove the existence of a complex structure on the six-sphere" sounds less self-assured than the title! – Mark Grant Jan 02 '11 at 13:25
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    I have not looked at this paper yet. Thanks for pointing it out. However, you should change the title of your question, since your question really is about a specific paper. (In fact, I suspect the question might get closed because these sorts of questions usually are.) In any case, more background in general about this problem is discussed here: http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere – Spiro Karigiannis Jan 02 '11 at 14:06
  • Thanks for your comments. I have edited my title in "A paper to the question, if the six dimensional sphere is a complex manifold". I hope, someone of the experts have read this paper... Best regards Florian M. – Florian Modler Jan 02 '11 at 16:21
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    Hi Florian. With no disrespect intended, I have voted to close because questions of the form "is this proof correct?" have been deemed inappropriate for MO. On the other hand, a question more like "I am having trouble understanding step X on line 24 of page 5 of " are considered less controversial, more direct, and thus acceptable. If I am mistaken concerning the etiquette, then I apologize in advance. – David Jordan Jan 02 '11 at 16:38
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    @David: I don't think the closure etiquette here is so clear. I agree that asking if one's own proof is correct is off-topic for MO. However, asking about a preprint could be okay. In this instance, I voted up the question because I learned something from it: someone has made an apparently very serious attempt to resolve the conjecture of a complex structure on $S^6$. As a nonexpert in the area, I'm not going to get far by reading the paper itself, so I'd like to know what others think. As long as this can be done in a respectful way, I think it is fine for MO. – Pete L. Clark Jan 02 '11 at 16:45
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    I suggest that the second paragraph be replaced by something like: "I'd like to know if anyone can explain why the proof is correct or why the approach used by the proof should work". – Deane Yang Jan 02 '11 at 19:18
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    I'm with Pete L Clark here: I usually upvote a question when I learn something from the comments/answers, even if, were I to consider the question in a void, I'd vote to close. – Dr Shello Jan 03 '11 at 00:52
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    There is now an updated (i.e., fourth) version of the paper in question: http://arxiv.org/pdf/math/0505634v4 – mathphysicist Jan 11 '11 at 00:22
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    Note that the paper was withdrawn 2 days after it was originally posted in 2005, before being resubmitted. The approach outlined in the new abstract is completely different to the old approach. This lessens my a priori confidence in the paper, but I'm not an expert in that area, and I haven't looked at it. I'd also like to know what others think. – David Roberts Jan 12 '11 at 10:42
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    I can't speak for others, but I personally prefer links to abstracts (perhaps with a note about version number) instead of pdfs. – S. Carnahan Jan 12 '11 at 15:28
  • A new version of the paper has been posted: http://arxiv.org/abs/math/0505634 – Joseph O'Rourke Jun 13 '11 at 13:23
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    Update: Recently, a revised version (now v7) has been published (Journal of Mathematical Physics 01/2014). The mathematical part looks readable now. However, the introduction still says "we hope to prove rigorously", and the main theorem starts with "Assume that the Principle in Sect. 2 holds.", where the mentioned principle (p. 8) still seems a little obscure to me. – Ben May 24 '15 at 17:00
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    So a 6-sphere both has and hasn't a complex manifold structure. Is this, at last, the contradiction in mathematics which many have long waited for? – Wojowu Oct 31 '16 at 15:09

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