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The homotopy groups of the \(L_ 2\)-localized Mahowald spectrum \(X\langle 1\rangle\). (English) Zbl 0835.55009

Summary: The Mahowald spectrum \(X \langle k\rangle\) is characterized by the \(BP_*\)-homology as \[ BP_* (X \langle k\rangle) = (BP_*/(2, v_1, \dots, v_{k - 1})) [t_1, \dots, t_k] \] at the prime 2. For the Bousfield localization \(L_k\) with respect to \(v^{-1}_k BP\), Ravenel computed the homotopy groups of \(L_k X \langle k\rangle\), in which case \(E_2 = E_\infty\) in the Adams-Novikov spectral sequence (cf. [D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure Appl. Math. 121, Academic Press (1986; Zbl 0608.55001)]). It will be interesting to know about the homotopy groups of \(L_k X \langle l \rangle\) for \(k > l\). In this paper, we determine the homotopy groups for the case \(k = 2\) and \(l = 1\) by computing \(E_5 = E_\infty\) in the Adams-Novikov spectral sequence.

MSC:

55Q52 Homotopy groups of special spaces
55T15 Adams spectral sequences
55P42 Stable homotopy theory, spectra
55P60 Localization and completion in homotopy theory
55Q40 Homotopy groups of spheres

Citations:

Zbl 0608.55001
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