K -THEORY AND THE HOPF INVARIANT
K -THEORY AND THE HOPF INVARIANT
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K 理论和 HOPF 不变量
DOI：
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发表时间：
1966
期刊：
影响因子：
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通讯作者：
M. Atiyah
中科院分区：
文献类型：
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作者：
J. F. Adams;M. Atiyah
在(1)中，利用二次上同运算，证明了对于n ^ 1、2、4、8，Hopf不变量1在ir%n-^(S)中的不存在性。本文的主要目的是展示如何在。K"理论为这个结果提供了一个极其简单的替代证明。事实上，在(8)和(4)中已经给出了理论证明，但这两个证明都不是初等的：(8)使用复配的结果，而(4)使用陈氏特征与(3)中建立的Steenrod平方之间的联系[但见(6)对(3)结果的更初等的处理]。我们目前的方法的简单性和新颖性在于，与以前对Hopf不变量问题的所有攻击不同，我们考虑的不是问题的稳定版本，而是问题的不稳定版本：也就是说我们将直接证明
[Received 29 December 1964] Introduction THE non-existence of elements of Hopf invariant one in ir%n-^(S ), for n ^ 1, 2, 4, or 8, was established in (1) by the use of secondary cohomology operations. The main purpose of this paper is to show how the use of primary operations in .K"-theory provides an extremely simple alternative proof of this result. In fact ^-theory proofs have already been given in (8) and (4) but neither of these proofs is elementary: (8) uses results on complex cobordism, while (4) uses the connexion between the Chern character and the Steenrod squares established in (3) [see however (6) for a more elementary treatment of the results of (3)]. The simplicity and novelty of our present approach is that, unlike all previous attacks on the Hopf invariant problem, we consider not the stable but the unstable version of the problem: that is to say we shall prove directly