## Journal of the Mathematical Society of Japan

### The homotopy groups of the $L_{2}$-localized mod 3 Moore spectrum

Katsumi SHIMOMURA

#### Abstract

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

#### Article information

Source
J. Math. Soc. Japan, Volume 52, Number 1 (2000), 65-90.

Dates
First available in Project Euclid: 10 June 2008

https://projecteuclid.org/euclid.jmsj/1213107656

Digital Object Identifier
doi:10.2969/jmsj/05210065

Mathematical Reviews number (MathSciNet)
MR1727130

Zentralblatt MATH identifier
0946.55006

#### Citation

SHIMOMURA, Katsumi. The homotopy groups of the $L_{2}$ -localized mod 3 Moore spectrum. J. Math. Soc. Japan 52 (2000), no. 1, 65--90. doi:10.2969/jmsj/05210065. https://projecteuclid.org/euclid.jmsj/1213107656