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

Yoshitaka Nakazawa; Katsumi Shimomura

Fundamenta Mathematicae (1997)

- Volume: 152, Issue: 1, page 1-20
- ISSN: 0016-2736

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topNakazawa, Yoshitaka, and Shimomura, Katsumi. "The homotopy groups of the L2 -localization of a certain type one finite complex at the prime 3." Fundamenta Mathematicae 152.1 (1997): 1-20. <http://eudml.org/doc/212196>.

@article{Nakazawa1997,

abstract = {For the Brown-Peterson spectrum BP at the prime 3, $v_2$ denotes Hazewinkel’s second polynomial generator of $BP_*$. Let $L_2$ denote the Bousfield localization functor with respect to $v_2^\{-1\}BP$. A typical example of type one finite spectra is the mod 3 Moore spectrum M. In this paper, we determine the homotopy groups $π_*(L_2M ∧ X)$ for the 8 skeleton X of BP.},

author = {Nakazawa, Yoshitaka, Shimomura, Katsumi},

journal = {Fundamenta Mathematicae},

keywords = {homotopy groups; Adams-Novikov spectral sequence; Johnson-Wilson spectrum; coalgebroid; duality; iterated Adams map},

language = {eng},

number = {1},

pages = {1-20},

title = {The homotopy groups of the L2 -localization of a certain type one finite complex at the prime 3},

url = {http://eudml.org/doc/212196},

volume = {152},

year = {1997},

}

TY - JOUR

AU - Nakazawa, Yoshitaka

AU - Shimomura, Katsumi

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

JO - Fundamenta Mathematicae

PY - 1997

VL - 152

IS - 1

SP - 1

EP - 20

AB - For the Brown-Peterson spectrum BP at the prime 3, $v_2$ denotes Hazewinkel’s second polynomial generator of $BP_*$. Let $L_2$ denote the Bousfield localization functor with respect to $v_2^{-1}BP$. A typical example of type one finite spectra is the mod 3 Moore spectrum M. In this paper, we determine the homotopy groups $π_*(L_2M ∧ X)$ for the 8 skeleton X of BP.

LA - eng

KW - homotopy groups; Adams-Novikov spectral sequence; Johnson-Wilson spectrum; coalgebroid; duality; iterated Adams map

UR - http://eudml.org/doc/212196

ER -

## References

top- [1] 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. Zbl0869.55010
- [2] 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. Zbl0857.55003
- [3] 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. Zbl0791.55005
- [4] H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in Adams-Novikov spectral sequence, Ann. of Math. 106 (1977), 469-516. Zbl0374.55022
- [5] D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, 1986.
- [6] D. C. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Ann. of Math. Stud. 128, Princeton Univ. Press, 1992.
- [7] K. Shimomura, On the Adams-Novikov spectral sequence and products of β-elements, Hiroshima Math. J. 16 (1986), 209-224. Zbl0605.55013
- [8] K. Shimomura, The homotopy groups of the ${L}_{2}$-localized Mahowald spectrum X⟨1⟩, Forum Math. 7 (1995), 685-707. Zbl0835.55009
- [9] 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. Zbl0869.55012
- [10] 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. Zbl0606.55009
- [11] K. Shimomura and A. Yabe, The homotopy groups ${\pi}_{*}\left({L}_{2}{S}^{0}\right)$, Topology 34 (1995), 261-289. Zbl0832.55011

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