-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

Published electronically:
June 2, 2004

MathSciNet review:
2063134

Full-text PDF Free Access

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.

**1.**M. Behrens and S. Pemmaraju, On the existence of the self-map on the Smith-Toda complex at the prime , to appear in the Proceedings of the Northwestern University Algebraic Topology Conference, March 2002.**2.**P. Goerss, H.-W. Henn and M. Mahowald, The homotopy of for the prime , to appear in the Proceedings of the International Conference on Algebraic Topology, the Isle of Skye, 2001.**3.**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**, 10.1090/conm/158/01454**4.**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**, 10.1112/S0024610799007784**5.**Y. Kamiya and K. Shimomura, A relation between the Picard group of the -local homotopy category and -based Adams spectral sequence, to appear in the Proceedings of the Northwestern University Algebraic Topology Conference, March 2002.**6.**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**, 10.1090/conm/314/05428**7.**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**0458423****8.**Shichirô Oka,*Ring spectra with few cells*, Japan. J. Math. (N.S.)**5**(1979), no. 1, 81–100. MR**614695****9.**Shichirô Oka,*Note on the 𝛽-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**, 10.2206/kyushumfs.35.367**10.**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****11.**Katsumi Shimomura,*The homotopy groups of the 𝐿₂-localized Toda-Smith spectrum 𝑉(1) at the prime 3*, Trans. Amer. Math. Soc.**349**(1997), no. 5, 1821–1850. MR**1370651**, 10.1090/S0002-9947-97-01710-8**12.**Katsumi Shimomura,*The homotopy groups of the 𝐿₂-localized mod 3 Moore spectrum*, J. Math. Soc. Japan**52**(2000), no. 1, 65–90. MR**1727130**, 10.2969/jmsj/05210065**13.**Katsumi Shimomura,*On the action of 𝛽₁ in the stable homotopy of spheres at the prime 3*, Hiroshima Math. J.**30**(2000), no. 2, 345–362. MR**1777519****14.**Katsumi Shimomura and Xiangjun Wang,*The homotopy groups 𝜋_{*}(𝐿₂𝑆⁰) at the prime 3*, Topology**41**(2002), no. 6, 1183–1198. MR**1923218**, 10.1016/S0040-9383(01)00033-7**15.**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**, 10.1017/CBO9780511526312.010**16.**Hirosi Toda,*Algebra of stable homotopy of 𝑍_{𝑝}-spaces and applications*, J. Math. Kyoto Univ.**11**(1971), 197–251. MR**0293631**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
55Q99,
55Q45,
55Q51

Retrieve articles in all journals with MSC (2000): 55Q99, 55Q45, 55Q51

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