-invertible spectra smashing with the Smith-Toda spectrum at the prime

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 | References | Similar Articles | Additional Information

Abstract: Let denote the Bousfield localization functor with respect to the Johnson-Wilson spectrum . A spectrum is called invertible if there is a spectrum such that . Hovey and Sadofsky, *Invertible spectra in the **-local stable homotopy category*, showed that every invertible spectrum is homotopy equivalent to a suspension of the -local sphere at a prime . At the prime , it is shown, *A relation between the Picard group of the **-local homotopy category and **-based Adams spectral sequence*, that there exists an invertible spectrum that is not homotopy equivalent to a suspension of . In this paper, we show the homotopy equivalence for the Smith-Toda spectrum . In the same manner as this, we also show the existence of the self-map that induces on the -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