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

Ichigi, Ippei; Shimomura, Katsumi

doi:10.1090/S0002-9939-04-07387-3

$E(2)$-invertible spectra smashing with the Smith-Toda spectrum $V(1)$ at the prime $3$
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by Ippei Ichigi and Katsumi Shimomura PDF

Proc. Amer. Math. Soc. 132 (2004), 3111-3119 Request permission

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.

References

M. Behrens and S. Pemmaraju, On the existence of the self-map $v_2^9$ on the Smith-Toda complex $V(1)$ at the prime $3$, to appear in the Proceedings of the Northwestern University Algebraic Topology Conference, March 2002.
P. Goerss, H.-W. Henn and M. Mahowald, The homotopy of $L_2V(1)$ for the prime $3$, to appear in the Proceedings of the International Conference on Algebraic Topology, the Isle of Skye, 2001.
Michael J. Hopkins, Mark Mahowald, and Hal Sadofsky, Constructions of elements in Picard groups, Topology and representation theory (Evanston, IL, 1992) Contemp. Math., vol. 158, Amer. Math. Soc., Providence, RI, 1994, pp. 89–126. MR 1263713, DOI 10.1090/conm/158/01454
Mark Hovey and Hal Sadofsky, Invertible spectra in the $E(n)$-local stable homotopy category, J. London Math. Soc. (2) 60 (1999), no. 1, 284–302. MR 1722151, DOI 10.1112/S0024610799007784
Y. Kamiya and K. Shimomura, A relation between the Picard group of the $E(n)$-local homotopy category and $E(n)$-based Adams spectral sequence, to appear in the Proceedings of the Northwestern University Algebraic Topology Conference, March 2002.
Yousuke Kamiya and Katsumi Shimomura, $E_*$-homology spheres for a connective spectrum $E$, Topology and geometry: commemorating SISTAG, Contemp. Math., vol. 314, Amer. Math. Soc., Providence, RI, 2002, pp. 153–159. MR 1941628, DOI 10.1090/conm/314/05428
Haynes R. Miller, Douglas C. Ravenel, and W. Stephen Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (2) 106 (1977), no. 3, 469–516. MR 458423, DOI 10.2307/1971064
Shichirô Oka, Ring spectra with few cells, Japan. J. Math. (N.S.) 5 (1979), no. 1, 81–100. MR 614695, DOI 10.4099/math1924.5.81
Shichirô Oka, Note on the $\beta$-family in stable homotopy of spheres at the prime $3$, Mem. Fac. Sci. Kyushu Univ. Ser. A 35 (1981), no. 2, 367–373. MR 628729, DOI 10.2206/kyushumfs.35.367
Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042
Katsumi Shimomura, The homotopy groups of the $L_2$-localized Toda-Smith spectrum $V(1)$ at the prime $3$, Trans. Amer. Math. Soc. 349 (1997), no. 5, 1821–1850. MR 1370651, DOI 10.1090/S0002-9947-97-01710-8
Katsumi Shimomura, The homotopy groups of the $L_2$-localized mod $3$ Moore spectrum, J. Math. Soc. Japan 52 (2000), no. 1, 65–90. MR 1727130, DOI 10.2969/jmsj/05210065
Katsumi Shimomura, On the action of $\beta _1$ in the stable homotopy of spheres at the prime 3, Hiroshima Math. J. 30 (2000), no. 2, 345–362. MR 1777519
Katsumi Shimomura and Xiangjun Wang, The homotopy groups $\pi _*(L_2S^0)$ at the prime 3, Topology 41 (2002), no. 6, 1183–1198. MR 1923218, DOI 10.1016/S0040-9383(01)00033-7
N. P. Strickland, On the $p$-adic interpolation of stable homotopy groups, Adams Memorial Symposium on Algebraic Topology, 2 (Manchester, 1990) London Math. Soc. Lecture Note Ser., vol. 176, Cambridge Univ. Press, Cambridge, 1992, pp. 45–54. MR 1232198, DOI 10.1017/CBO9780511526312.010
Hirosi Toda, Algebra of stable homotopy of $Z_{p}$-spaces and applications, J. Math. Kyoto Univ. 11 (1971), 197–251. MR 293631, DOI 10.1215/kjm/1250523647