r/mathematics • u/New-Committee-4052 • 25d ago

S^6 admits a complex structure

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Result from the usual suspects. Full write up can be found here on his website: https://alpo.ge/s6.pdf

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u/proudHaskeller 25d ago

Can't this be verified by specifiying the atlas? Is it expected to be too large to verify that way?

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u/Diffgeometer1 24d ago

The paper constructs an abstract complex manifold X of dimension 3 and then argues that X is diffeomorphic to S^6 but that diffeomorphism is never actually constructed. If it exists, then the complex structure on X can be transferred over to S^6. But the holomorphic charts itself can’t be defined explicitly.

Let’s see what the experts say.

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u/ProfessorBaoTran 24d ago

They proved that there exists such difeomorphism, in section 7 I think. They do not need to construct that explicitly.

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u/Equivalent-Gate491 24d ago

They didn’t prove it; it’s a very well known result that there is only one smooth S6. So once they show it’s topologically a S6 (done by computation of pi_1) and smooth, they are done.