## Journal of the Mathematical Society of Japan

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

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

#### Abstract

At each prime number $p$, the homotopy groups ${\pi}_{*}\left({L}_{2}{S}^{0}\right)$ of the ${{v}_{2}}^{-}$*B**P*-localized sphere spectrum play an crucial role to understand the category of ${{v}_{2}}^{-}$*B**P*-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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 10 June 2008

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/jmsj/05210065

**Mathematical Reviews number (MathSciNet)**

MR1727130

**Zentralblatt MATH identifier**

0946.55006

**Subjects**

Primary: 55Q45: Stable homotopy of spheres

Secondary: 55T15: Adams spectral sequences 55Q52: Homotopy groups of special spaces 55P42: Stable homotopy theory, spectra

**Keywords**

Homotopy groups Adams-Novikov spectral sequence Bousfield-Ravenel localization

#### 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