Geodesic
On R L Cohen’s ζ–element
Liu, Xiugui1
1School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
Algebraic and Geometric Topology, Tome 11 (2011) no. 3, pp. 1709-1735
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Let p be a prime greater than three. In the p–local stable homotopy groups of spheres, R L Cohen constructed the infinite ζ–element ζn−1 ∈ π2pn+1−2pn+2p−5(S) of order p. In the stable homotopy group π2pn+1−2pn+2p2−3(V (1)) of the Smith–Toda spectrum V (1), X Liu constructed an essential element ϖk for k ≥ 3. Let βs∗ = j0j1βs ∈ [V (1),S]2sp2−2s−2p denote the Spanier–Whitehead dual of the generator βs′′ = βsi1i0 ∈ π2sp2−2s(V (1)), which defines the β–element βs. Let ξs,k = βs−1∗ϖk. In this paper, we show that the composite of R L Cohen’s ζ–element ζn−1 with ξs,n is nontrivial, where n > 4 and 1 < s < p − 1. As a corollary, ξs,n is also nontrivial for 1 < s < p − 1.

DOI : 10.2140/agt.2011.11.1709
Keywords: stable homotopy groups of spheres, $\zeta$–element, Adams spectral sequence, May spectral sequence

Liu, Xiugui  1

1 School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China 
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Liu, Xiugui. On R L Cohen’s ζ–element. Algebraic and Geometric Topology, Tome 11 (2011) no. 3, pp. 1709-1735. doi: 10.2140/agt.2011.11.1709

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