The Smith-Toda complex $V((p+1)/2)$ does not exist
The Smith-Toda complex $V((p+1)/2)$ does not exist
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DOI：
10.4007/annals.2010.171.491
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发表时间：
2010-03
影响因子：
4.9
通讯作者：
Lee S. Nave
中科院分区：
文献类型：
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作者：
Lee S. Nave
代数拓扑学最重要的现代发展之一是在稳定同伦中发现了周期现象。这始于亚当斯关于J同态的工作[Ada66]，后来由L.Smith[Smi70]和H.Toda[Tod71]继续。设p是素数，V(0)k表示S上p次映射的余纤维，Adams和Toda证明了对于足够大的k，存在一个非幂零映射ΣV(0)kα−→V(0)k，其中q=2(p−1)，如果p是奇数，则q=8.设α表示合成ΣV(0)k−→··−→ΣV(0)kΣα−→ΣV(0)kα−→V(0)k，那么我们用非幂零表示没有α是零同伦的.通过包含ΣV(0)k的底胞并投影到V(0)k的顶胞，我们得到了球面同伦中的一族元素，也记为αt：S−→ΣV(0)kα−→V(0)k−→S.
One of the most significant modern developments in algebraic topology has been the discovery of periodic phenomena in stable homotopy. This began with Adams’ work on the J homomorphism [Ada66] and was later continued by L. Smith [Smi70] and H. Toda [Tod71]. Let p be a prime and let V (0)k denote the cofiber of the degree p map on S. Adams and Toda showed that for k sufficiently large there is a nonnilpotent map ΣV (0)k α −→ V (0)k, where q = 2(p− 1) if p is odd and q = 8 if p = 2. Let α denote the composite ΣV (0)k −→ · · · −→ ΣV (0)k Σα −→ ΣV (0)k α −→ V (0)k. Then by nonnilpotent, we mean that no α is nullhomotopic. By including the bottom cell of ΣV (0)k and projecting to the top cell of V (0)k, we obtain a family of elements, also denoted αt, in the homotopy of spheres: S −→ ΣV (0)k α −→ V (0)k −→ S.