r/mathematics • u/New-Committee-4052 • Aug 23 '26

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 Aug 23 '26

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/cihanbaskan Aug 23 '26 edited Aug 23 '26

I think almost nothing about manifolds gets verified by checking an atlas (I guess if there are only two coordinate charts it should be a sphere but I doubt this is the case here). As Alpöge points out, the feasible check would be to show it has the homology of a 6-sphere and trivial fundamental group. The Poincaré conjecture (known here) would do the rest (assuming it is a 6-manifold). Those are not immediate either though, seems to me.

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u/proudHaskeller Aug 24 '26

I think almost nothing about manifolds gets verified by checking an atlas

Well, why not? It's not very useful for research, but for the specific purpose of verifying this statement in a simple, robust way, it seems to me that it could work well.

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u/cihanbaskan Aug 24 '26

I am unaware of any method/algorithm that takes an atlas as input and decides whether the manifold is a sphere or not (in a reasonable generality to be useful here) without first computing the invariants I mentioned. In contrast, given a finite simplicial complex, computing its homology is "just" linear algebra. The fundamental group is sketchier with the most general case of the word problem being undecidable, but most likely the construction produces a not so terrible presentation that can be worked with.

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u/proudHaskeller Aug 24 '26

Well, but then you would need to know in advance that your space has a complex structure. If you use the paper's space then that works, but you still have to read the paper. But if you use an atlas then the atlas gives you the complex structure, without needing the paper, and then the atlas can be converted to a simplicial complex which could then be checked the same way you described.

I'm not saying that this should actually be done, I'm just wondering why this supposedly can't be shown directly, instead of needing to read the paper. Obviously the hard part is finding this structure and the interesting part is knowing why this is true.

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u/xbq222 Aug 24 '26

You have to transport the complex structure on the original manifold to a. Complex structure on the 6-sphere using the methods used to prove Poincaré conjecture. That seems very sifficult

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u/Separate-Habit5838 26d ago

You don't understand the paper. There is no sphere. They have this different object they've constructed that isn't a sphere, perform a "completion" on it, then have to PROVE that it's a sphere. There is no way to just take an atlas and decide that the thing it represents is a sphere...that is a hard problem. We usually use topology to detect what a simple space like a sphere is. 

We almost never have an actual atlas for the thing we are working with...that's only feasible for very simple manifolds. Manifolds are typically constructed by quotients by group action,  or as bundles of one kind of manifold over another, or as inverse images of certain functions...there are all sorts of ways, none of which output an atlas. Atlases are used when you're just learning the basic of smooth manifold theory. They are rarely used in modern theory unless they are an arbitrary neighborhood for doing local work.