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$E(2)$-invertible spectra smashing with the Smith-Toda spectrum $V(1)$ at the prime $3$


Authors: Ippei Ichigi and Katsumi Shimomura
Journal: Proc. Amer. Math. Soc. 132 (2004), 3111-3119
MSC (2000): Primary 55Q99; Secondary 55Q45, 55Q51
DOI: https://doi.org/10.1090/S0002-9939-04-07387-3
Published electronically: June 2, 2004
MathSciNet review: 2063134
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Abstract: Let $L_2$ denote the Bousfield localization functor with respect to the Johnson-Wilson spectrum $E(2)$. A spectrum $L_2X$ is called invertible if there is a spectrum $Y$ such that $L_2X\wedge Y=L_2S^0$. Hovey and Sadofsky, Invertible spectra in the $E(n)$-local stable homotopy category, showed that every invertible spectrum is homotopy equivalent to a suspension of the $E(2)$-local sphere $L_2S^0$ at a prime $p>3$. At the prime $3$, it is shown, A relation between the Picard group of the $E(n)$-local homotopy category and $E(n)$-based Adams spectral sequence, that there exists an invertible spectrum $X$ that is not homotopy equivalent to a suspension of $L_2S^0$. In this paper, we show the homotopy equivalence $v_2^3\colon \Sigma^{48}L_2V(1)\simeq V(1)\wedge X$ for the Smith-Toda spectrum $V(1)$. In the same manner as this, we also show the existence of the self-map $\beta\colon \Sigma^{144}L_2V(1)\to L_2V(1)$ that induces $v_2^9$ on the $E(2)_*$-homology.


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Additional Information

Ippei Ichigi
Affiliation: Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan
Email: 95sm004@math.kochi-u.ac.jp

Katsumi Shimomura
Affiliation: Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan
Email: katsumi@math.kochi-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-04-07387-3
Keywords: Invertible spectrum, Smith-Toda spectrum, homotopy groups
Received by editor(s): November 20, 2002
Received by editor(s) in revised form: May 23, 2003
Published electronically: June 2, 2004
Communicated by: Paul Goerss
Article copyright: © Copyright 2004 American Mathematical Society

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