The homotopy groups of the L2 -localization of a certain type one finite complex at the prime 3

ArticleOriginal scientific text

Title

Authors ¹, ²

Yoshitaka Nakazawa, 1-22-7, Omokage, Tottori, 680, Japan
Katsumi Shimomura, Faculty of Education, Tottori University, Tottori, 680, Japan

For the Brown-Peterson spectrum BP at the prime 3,

v_{2}

denotes Hazewinkel's second polynomial generator of

B P_{\cdot}

. Let

L_{2}

denote the Bousfield localization functor with respect to

v_{2}^{- 1} B P

. A typical example of type one finite spectra is the mod 3 Moore spectrum M. In this paper, we determine the homotopy groups

π_{\cdot} (L_{2} M \land X)

for the 8 skeleton X of BP.

Y. Arita and K. Shimomura, The chromatic $E_{1}$ -term $H^{1} M_{1}^{1}$ at the prime 3, Hiroshima Math. J. 26 (1996), 415-431.
M. J. Hopkins and B. H. Gross, The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc. 30 (1994), 76-86.
M. Mahowald and K. Shimomura, The Adams-Novikov spectral sequence for the $L_{2}$ -localization of a $v_{2}$ -spectrum, in: Contemp. Math. 146, Amer. Math. Soc., 1993, 237-250.
H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in Adams-Novikov spectral sequence, Ann. of Math. 106 (1977), 469-516.
D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, 1986.
D. C. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Ann. of Math. Stud. 128, Princeton Univ. Press, 1992.
K. Shimomura, On the Adams-Novikov spectral sequence and products of β-elements, Hiroshima Math. J. 16 (1986), 209-224.
K. Shimomura, The homotopy groups of the $L_{2}$ -localized Mahowald spectrum X⟨1⟩, Forum Math. 7 (1995), 685-707.
K. Shimomura, The homotopy groups of the $L_{2}$ -localized Toda-Smith spectrum V(1) at the prime 3, Trans. Amer. Math. Soc., to appear.
K. Shimomura and H. Tamura, Non-triviality of some compositions of β-elements in the stable homotopy of Moore spaces, Hiroshima Math. J. 16 (1986), 121-133.
K. Shimomura and A. Yabe, The homotopy groups $π_{\cdot} (L_{2} S^{0})$ , Topology 34 (1995), 261-289.