An important relation in homotopy groups of spheres
An important relation in homotopy groups of spheres
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球面同伦群中的重要关系
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发表时间：
1967
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通讯作者：
H. Toda
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作者：
H. Toda
1.在计算k-茎群G-lim +（S）的p-准素分支G时，p总是表示奇素数，一个本质的困难在于k-2 p（p-1）-3的情形。最近，科恩1已经宣布，G，.，__ Z_i，这等价于说G_i和G_i的生成元为0。然而，目前的工作结果并不同意这一宣布。我们的基本结果是定理。对任意大的整数n，存在胞腔复形g Se+（-'（-' U e+（-'e+（-））使得H（K; Z）O，2 H+（-）（-）-（K; Z）O，且胞腔e+（-'（-'）通过代表连接到S上.它直接遵循以下推论。f-0这表明，在5月2日计算的亚当斯谱序列中，与B相消的微分消元，或等价地，G（_）_的元素不存在，应与f相消。因此，G的修正结果如下：命题1。对于k2（p +2 p）（p1）4，G是由对应次数k”（1gi <p+2 p，i 0（mod p）），（lgj<p+ 2，j 0（rood p）），“（lr<p+3），a（lr<p），p，
1. In computing the p-primary component G of the k-stem group G-lim +(S), p denoting always an odd prime, an essential difficulty lies in the case k-2p(p-1)-3. Recently, Cohen 1 has announced that G,.,__Z,, which is equivalent to say that 0 for the generators of G_ and of G__. The result of the present work, however, does not agree with this announcement. Our fundamental result is Theorem. For suciently large integer n, there exists a cell complex g S e+(-’(-’U e+(-’e+(-) such that H(K; Z)O, 2H+(-)(-)-(K; Z)0 and the cell e+(-’(-’is attached to S by a representative of . It follows immediately the following Corollary. f-0. This shows that, in the Adams’ spectral sequence computed by May 2, the differential cancells hob with b, or equivalently, the element of G(_)_ does not exist and should be cancelled with f. Then the corrected results for G are stated as follows: Proposition 1. For k 2(p + 2p)(p1) 4, G is the direct sum of cyclic groups generated by the following elements of corresponding degree k" (1g i<p+2p, i 0(mod p)), (lgj<p+ 2, j 0(rood p)), " (lr<p+3), a(lr<p), p,