如何看待 Atiyah 对六维球面 S^6 上没有复结构的证明？ 第1页

谢邀。

我基本没怎么学过K理论，今天刚和老板讨论这个问题，他代数拓扑知道得比我多，但是对指标理论和K理论也不怎么了解。不过他觉得基本不太可能是对的，因为真正的“证明过程”只有半页纸，而且看看最后的致谢部分怎么写的：“Dennis Sullivan and Claude Lebrun showed some healthy and fruitful scepticism. My younger colleagues, Nigel Hitchin and Ju ̈rgen Berndt, provided expertise and constructive criticism. Robert Bryant who knows the problem well [13] pushed me hard and, supported by Blaine Lawson, forced me to clarify my argument. Finally my amanu- ensis Andrew Ranicki has always been at my elbow, speeding me along.” 我老板的说法是，一个数学家，如果他已经87岁了，被这么多大人物和领域内专家质疑，他是应该要认真听听的。。PS:

这个致谢根本就不是“致谢里说那么多人检查过了也不会有错”，而是这么多人检查过了纷纷表示质疑啊。。

我个人还是希望他真做出来了的，毕竟学数学的都会有点英雄主义，心里还是期望一代宗师能够宝刀不老的。不过数学界是讲论证的地方，不是讲感情的地方，就目前情况来说，数学界还没有公认他的结果，很多人都表示质疑，还是先看一段时间热闹比较好。如果真如

所说，证明只是本科生大作业程度的话，那这么多专家质疑也太不正常了。。

更新：证明过程主要在section 3,所以我们来看看section 3讲了些什么：

3. Almost complex structures

Let us begin by examining the almost complex structure J(0) on the 6-sphere induced by the octonions. It is immaterial whether we take the Euclidean signature or the Minkowski one, but we prefer the Minkowski one with signature (7,1).In fact we have not one almost complex struc- ture (ACS) but two conjugate ones. Just as in note 3, we should not distinguish between them, the pair come as a single real ACS. Locally, if we puncture the 6-sphere at one point, we can distinguish them, calling one + and one -, but this distinction is not necessarily global. There are topologically just two possibilities, either the distinction is global (the “even” case) and carries across the puncture or it is not (the “odd” case).

这个自然段是说S^6上已知的两个（共轭的）近复结构不可积，但这并不意味着什么，因为这是几十年前就被证明的结果了。然后这里提出了even和odd的区别，主要区别似乎是这两个近复结构的区分是不是整体的。

Whether we are in the even case or the odd case depends in general on the situation. For the 6-sphere, the base of the light cone in Minkowski space R(7, 1), we are in the odd case, but for signatures (5,3) or dually (3,5), the 6-sphere is replaced by the complex projective 3-space P, or its dual P∗, the real ACS is integrable and we are in the even case. This is just linear algebra and easily checked.（我感觉这个“容易验证的线性代数”写出来肯定不止7页。。）

The linear algebra involved is precisely that which enters into triality for the spinor group of signature (7,1), and the associated signatures (5,3) and (3,5).

这一段我看不懂，学识太有限没办法。。黑体部分我感觉是他的论据之一？但是也只是个claim，并没有证明啊。。

Because of the Atiyah-Singer theory, the linear algebra acquires a topological meaning, and that is embodied in KR theory.

这也讲得太简略了吧。。

Our Theorem can now be reinterpreted as saying that, on the 6- sphere, any real ACS is of odd type. Hence there is no real ACS of even type. An integrable complex structure would define a real ACS of even type and so cannot exist. The integrability condition is essentially replaced by an equivalent topological condition.（为毛？） This for the 6-sphere is precisely what we expected, since there is no local obstruction and we needed a global cohomological obstruction in an appropriate theory.（我们需要一个global cohomological obstruction，所以这个obstruction就自动出现了么？你倒是说说这个global cohomological obstruction到底是啥啊。。） That theory is just KR theory. A short history of the 6-sphere problem follows as section 4.

然后接下来section 4，他又开始扯与证明无关的历史了。。