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Some infinite elements in the Adams spectral sequence for the sphere spectrum. (English) Zbl 1208.55008

Summary: In the stable homotopy groups \(\pi_{p^nq+(p+1)q-1}(V(1))\) of the Smith-Toda spectrum \(V(1)\), the author constructs an essential element \(\varpi_n\) for \(n\geq 3\) at the prime greater than three. Let \(\beta^*_s\in [V(1),S]_{spq+(s-1)q-2}\) denote the dual of the generator \(\beta''_s\in \pi_{s(p+1)q}(V(1))\), which defines the \(\beta\)-element \(\beta_s\). In this paper, the author shows that the composite \(\alpha_1\beta_1\xi_s\in\pi_{p^nq+(s+1)pq+sq-6}(S)\) for \(1<s<p-2\) is non-trivial, where \(\xi_s=\beta^*_{s-1}\varpi_n\in\pi_{p^nq+spq+(s-1)q-3}(S)\) and \(q=2(p-1)\). As a corollary, \(\xi_s\), \(\alpha_1\xi_s\) and \(\beta_1\xi_s\) are also non-trivial for \(1<s<p-2\).

MSC:

55Q45 Stable homotopy of spheres
55T15 Adams spectral sequences
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