Abstract: Let
L2denote
the Bousfield localization functor with respect
to the second Johnson-Wilson spectrum E(2).A spectrum L2X is
called invertible if there is a spectrum Y
such that L2XÙY = L2S0.
Then
Hovey and Sadofsky showed that every invertible
spectrum is a suspension of the sphere spectrum L2S0if
the prime p is greater than three. At the prime three, Kamiya and the second
author constructed an invertible spectrum P
other than a suspension of L2S0,
and
showed a possibility of existence of another
invertible spectrum Q
such that every invertible spectrum has a
form åk
PÙpÙQÙqfor
integers kÎZand
p,
qÎZ/3,where
XÙXÙX = L2S0for
X
= P, Q.In
this paper, we consider the homotopy groups of
the invertible spectrum Q
under the assumption that Q
exists, and determine the homotopy groups p*(QÙqÙV(1))and
p*(PÙpÙQÙqÙV(1))for
the Smith-Toda spectrum V(1).
The results make the authors conjecture that Q
does not exist.
Keywords and phrases: homotopy group, invertible spectrum, Johnson-Wilson spectrum.