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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

marked as duplicate by Francois Ziegler, Wolfgang, András Bátkai, Willie Wong, Andy Putman Oct 31 '16 at 16:16

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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: mathoverflow.net/questions/1973/… – Spiro Karigiannis Jan 2 '11 at 14:06
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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 <arXiv link>" are considered less controversial, more direct, and thus acceptable. If I am mistaken concerning the etiquette, then I apologize in advance. – David Jordan Jan 2 '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 2 '11 at 16:45
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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
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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

Another paper has now been posted by Atiyah claiming to prove that there is no complex six sphere.

https://arxiv.org/abs/1610.09366

  • If only the question on major advances made by people more than fifty years old were still open, this should qualify as the first answer. – user57888 Oct 31 '16 at 11:37
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    Atiyah is much older than fifty. – Lubin Oct 31 '16 at 15:27
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    There appears to be serious doubt about the correctness of Atiyah's paper. – Jim Conant Oct 31 '16 at 16:18
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    I'm not an expert, but I did consult a couple. They confirmed that there are doubts about Atiyah's paper. – Deane Yang Oct 31 '16 at 18:38
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    @DavidRoberts, yes it was a private discussion. – Jim Conant Oct 31 '16 at 20:33

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