The homotopy groups of the L2-localized mod 3 Moore spectrum

Abstract

At each prime number $p$, the homotopy groups ${\pi }_{*}\left(L_2{S}^0\right)$ of the $v_2^{-1}$BP-localized sphere spectrum play an 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 the prime 3, ${\pi }_{*}\left(L_{2}V\right(1\left)\right)$ is also determined by using the ANSS, in which $E_{\infty }=$$E_{10}$ in this case. Here $V\left(1\right)$ denotes the Toda-Smith 4-cells spectrum. In this paper, we determine the homotopy groups ${\pi }_{*}\left(L_{2}V\right(0\left)\right)$ of the mod 3 Moore spectrum from ${\pi }_{*}$$(L_{2}V$(1 )$)$ 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.