The homotopy groups of the 𝐿₂-localized Toda-Smith spectrum 𝑉(1) at the prime 3

Abstract

In this paper, we try to compute the homotopy groups of the L 2 L_2 -localized Toda-Smith spectrum V ( 1 ) V(1) at the prime 3 by using the Adams-Novikov spectral sequence, and have almost done so. This computation involves non-trivial differentials d 5 d_5 and d 9 d_9 of the Adams-Novikov spectral sequence, different from the case p > 3 p>3 . We also determine the homotopy groups of some L 2 L_2 -localized finite spectra relating to V ( 1 ) V(1) . We further show some of the non-trivial differentials on elements relating so-called β \beta -elements in the Adams-Novikov spectral sequence for π ∗ ( S 0 ) \pi _*(S^0) .