Periodic phenomena in the Adams-Novikov spectral sequence
Periodic phenomena in the Adams-Novikov spectral sequence
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DOI：
10.2307/1971064
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发表时间：
1977-11
影响因子：
4.9
通讯作者：
H. Miller;D. Ravenel;W. Wilson
中科院分区：
文献类型：
--
作者：
H. Miller;D. Ravenel;W. Wilson
稳定同伦环的理解问题一直是代数拓扑学的试金石之一。低维计算进展缓慢，几乎没有深入了解7WS(S0)的一般结构。然而，近年来，7Rs(S0)的元素的无限族被发现，推广了Whitehead J-同态的形象。在这项工作中，我们提出了一个检测和描述这种无限族中的元素的通用程序。这种方法表明，在某种弱化意义上，每个同伦类都是这样一个族的成员。为了我们对同伦理论的代数把握，我们将使用S·P·诺维科夫对收敛到稳定同伦环的亚当斯谱序列的模拟。它的E_2项在代数上可以描述为复余弦中稳定运算的Landweber-Novikov代数的上同调。在他关于这个主题的开创性工作中，Novikov计算了第一个上同调群，并证明了它典型地同构于远离素数2的J的像。当定位在奇素数p时，这些元素每隔2(P1)维才出现一次；所以第一个上同调群具有周期性。我们在这里的目的是要证明，整个上同调是以一种非常特殊的方式从周期成分建立起来的。这些思想的主要应用是计算奇素数的第二上同调群。借助于Adams-Novikov谱序列，该信息具有许多同伦理论结果。本文研究了L.Smith构造的p>；3的(2(P21)t2(P1)2)-茎的p-分支中的同伦类ST，t>；1。事实上，事实证明，所有亚当斯-诺维科夫滤子恰好为2的元素都与这个家族密切相关。除了FI家族本身之外，过滤2的最低维度元素是由Toda表示的元素Ej。最小二乘法
The problem of understanding the stable homotopy ring has long been one of the touchstones of algebraic topology. Low dimensional computation has proceeded slowly and has given little insight into the general structure of 7ws(S0). In recent years, however, infinite families of elements of 7rs (S0) have been discovered, generalizing the image of the Whitehead J-homomorphism. In this work we indicate a general program for the detection and description of elements lying in such infinite families. This approach shows that every homotopy class is, in some attenuated sense, a member of such a family. For our algebraic grip on homotopy theory we shall employ S. P. Novikov's analogue of the Adams spectral sequence converging to the stable homotopy ring. Its E2-term can be described algebraically as the cohomology of the Landweber-Novikov algebra of stable operations in complex cobordism. In his seminal work on the subject, Novikov computed the first cohomology group and showed that it was canonically isomorphic to the image of J away from the prime 2. When localized at an odd prime p these elements occur only every 2(p 1) dimensions; so this first cohomology group has a periodic character. Our intention here is to show that the entire cohomology is built up in a very specific way from periodic constituents. Our central application of these ideas is the computation of the second cohomology group at odd primes. By virtue of the Adams-Novikov spectral sequence this information has a number of homotopy-theoretic consequences. The homotopy classes St, t > 1, in the p-component of the (2(p2 1)t 2(p 1) 2)-stem for p > 3, constructed by L. Smith, are detected here. Indeed, it turns out that all elements with Adams-Novikov filtration exactly 2 are closely related to the , family. The lowest dimensional elements of filtration 2 aside from the fi family itself are the elements denoted ej by Toda. The computation of the