The homotopy groups of the \(L_2\)-localized mod 3 Moore spectrum. (English) Zbl 0946.55006

At each prime number \(p\), the homotopy groups \(\pi_* (L_2S^0)\) of the \(v_2^{-1} BP\)-localized sphere spectrum play a crucial role to understand the category of \(v_2^{-1} BP\)-local spectra. For \(p>3\), they are determined by using the Adams-Novikov spectral sequence (ANSS), which collapses in this case. At a prime 3, \(\pi_*(L_2V(1))\) is also determined by using the ANSS, in which \(E_\infty= E_{10}\) in this case. Here \(V(1)\) denotes the Toda-Smith 4-cells spectrum. In this paper, we determine the homotopy groups \(\pi_*(L_2 M_3)\) of the mod 3 Moore spectrum from \(\pi_* (L_2V(1))\) by the Bockstein spectral sequence (BSS). Actually, we first compute the \(E_2\)-term of the ANSS by the BSS and then study the Adams-Novikov differentials, and obtain \(E_\infty=E_{10}\) as well.


55Q45 Stable homotopy of spheres
55T15 Adams spectral sequences
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55Q52 Homotopy groups of special spaces
55P42 Stable homotopy theory, spectra
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