The homotopy groups of at an odd prime
Authors:
Liu Xiugui, Wang Xiangjun and Yuan Zihong
Journal:
Proc. Amer. Math. Soc. 138 (2010), 11431152
MSC (2000):
Primary 55Q99, 55Q52
Published electronically:
October 28, 2009
MathSciNet review:
2566579
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper, all spectra are localized at an odd prime. Let be the Ravenel spectrum characterized by homology as , be the cofiber of the selfmap and denote the Bousfield localization functor with respect to . In this paper, we determine the homotopy groups of .
 1.
Mark
Hovey and Neil
P. Strickland, Morava 𝐾theories and localisation,
Mem. Amer. Math. Soc. 139 (1999), no. 666, viii+100.
MR
1601906 (99b:55017), 10.1090/memo/0666
 2.
Mark
Hovey and Neil
Strickland, Comodules and Landweber exact homology theories,
Adv. Math. 192 (2005), no. 2, 427–456. MR 2128706
(2006e:55007), 10.1016/j.aim.2004.04.011
 3.
Ippei
Ichigi, Katsumi
Shimomura, and Xiangjun
Wang, On subgroups of
𝜋_{∗}(𝐿₂𝑇(1)∧𝑀(2)) at the
prime two, Bol. Soc. Mat. Mexicana (3) 13 (2007),
no. 1, 207–230. MR 2468037
(2010d:55016)
 4.
Yousuke
Kamiya and Katsumi
Shimomura, The homotopy groups
𝜋_{*}(𝐿₂𝑉(0)∧𝑇(𝑘)),
Hiroshima Math. J. 31 (2001), no. 3, 391–408.
MR
1870983 (2002j:55010)
 5.
Haynes
R. Miller, Douglas
C. Ravenel, and W.
Stephen Wilson, Periodic phenomena in the AdamsNovikov spectral
sequence, Ann. of Math. (2) 106 (1977), no. 3,
469–516. MR 0458423
(56 #16626)
 6.
Hirofumi
Nakai and Katsumi
Shimomura, On the homotopy groups of 𝐸(𝑛)local
spectra with unusual invariant ideals, Proceedings of the Nishida Fest
(Kinosaki 2003), Geom. Topol. Monogr., vol. 10, Geom. Topol. Publ.,
Coventry, 2007, pp. 319–332. MR 2402792
(2009e:55021), 10.2140/gtm.2007.10.319
 7.
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
(87j:55003)
 8.
Douglas
C. Ravenel, Localization with respect to certain periodic homology
theories, Amer. J. Math. 106 (1984), no. 2,
351–414. MR
737778 (85k:55009), 10.2307/2374308
 9.
Douglas
C. Ravenel, Nilpotence and periodicity in stable homotopy
theory, Annals of Mathematics Studies, vol. 128, Princeton
University Press, Princeton, NJ, 1992. Appendix C by Jeff Smith. MR 1192553
(94b:55015)
 10.
Douglas
C. Ravenel, The cohomology of the Morava stabilizer algebras,
Math. Z. 152 (1977), no. 3, 287–297. MR 0431168
(55 #4170)
 11.
Douglas
C. Ravenel, The method of infinite descent in stable homotopy
theory. I, Recent progress in homotopy theory (Baltimore, MD, 2000)
Contemp. Math., vol. 293, Amer. Math. Soc., Providence, RI, 2002,
pp. 251–284. MR 1890739
(2003f:55009), 10.1090/conm/293/04951
 12.
Katsumi
Shimomura, The homotopy groups of the 𝐿₂localized
Mahowald spectrum 𝑋⟨1⟩, Forum Math.
7 (1995), no. 6, 685–707. MR 1359422
(96m:55023), 10.1515/form.1995.7.685
 13.
Katsumi
Shimomura and Atsuko
Yabe, The homotopy groups
𝜋_{*}(𝐿₂𝑆⁰), Topology
34 (1995), no. 2, 261–289. MR 1318877
(96b:55015), 10.1016/00409383(94)00032G
 14.
Katsumi
Shimomura and Xiangjun
Wang, The homotopy groups
𝜋_{*}(𝐿₂𝑆⁰) at the prime 3,
Topology 41 (2002), no. 6, 1183–1198. MR 1923218
(2003g:55020), 10.1016/S00409383(01)000337
 15.
Katsumi
Shimomura and Xiangjun
Wang, The AdamsNovikov 𝐸₂term for
𝜋_{*}(𝐿₂𝑆⁰) at the prime 2,
Math. Z. 241 (2002), no. 2, 271–311. MR 1935487
(2003m:55018), 10.1007/s002090200415
 16.
Xiangjun
Wang,
𝜋_{*}(𝐿₂𝑇(1)/(𝑣₁)) and
its applications in computing 𝜋_{*}(𝐿₂𝑇(1))
at the prime two, Forum Math. 19 (2007), no. 1,
127–147. MR 2296069
(2008a:55010), 10.1515/FORUM.2007.006
 1.
 Hovey, M. and Strickland N., Morava theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666. MR 1601906 (99b:55017)
 2.
 Hovey, M. and Strickland N., Comodules and Landweber exact homology theories, Adv. Math. 192 (2005), 427456. MR 2128706 (2006e:55007)
 3.
 Ichigi, I., Shimomura, K. and Wang, X., On subgroups of at the prime two, Bol. Soc. Mat. Mexicana (3) 13 (2007), 207230. MR 2468037
 4.
 Kamiya, Y. and Shimomura, K., The homotopy groups , Hiroshima Math. J. 31 (2001), 391408. MR 1870983 (2002j:55010)
 5.
 Miller, H., Ravenel, D. C. and Wilson, S., Periodic phenomena in the AdamsNovikov spectral sequence, Ann. of Math. (2) 106 (1977), 469516. MR 0458423 (56:16626)
 6.
 Nakai, H. and Shimomura, K., On the homotopy groups of local spactra with unusual invariant ideals, Geometry and Topology Monographs 10 (2007), 319332. MR 2402792 (2009e:55021)
 7.
 Ravenel, D. C., Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press (1986). MR 860042 (87j:55003)
 8.
 Ravenel, D. C., Localization with respect to certain periodic homology theories, American Journal of Mathematics 106 (1984), 351414. MR 737778 (85k:55009)
 9.
 Ravenel, D. C., Nilpotence and Periodicity in Stable Homotopy Theory, Princeton University Press (1992). MR 1192553 (94b:55015)
 10.
 Ravenel, D. C., The cohomology of the Morava stabilizer algebras, Math. Z. 152 (1977), 287297. MR 0431168 (55:4170)
 11.
 Ravenel, D. C., The method of infinite descent in stable homotopy theory. I, Contemporary Mathematics, 293, Amer. Math. Soc., Providence, RI, 2002, 251284. MR 1890739 (2003f:55009)
 12.
 Shimomura, K., The homotopy groups of the localized Mahowald spectrum , Forum Math. 7 (1995), 685707. MR 1359422 (96m:55023)
 13.
 Shimomura, K. and Yabe, A., The homotopy groups , Topology 34 (1995), 261289. MR 1318877 (96b:55015)
 14.
 Shimomura, K. and Wang, X., The homotopy groups of at the prime 3, Topology 41 (2002), 11831198. MR 1923218 (2003g:55020)
 15.
 Shimomura, K. and Wang, X., The AdamsNovikov term for at the prime 2, Math. Z. 241 (2002), 271311. MR 1935487 (2003m:55018)
 16.
 Wang, X., and its applications in computing at the prime two, Forum Math. 19 (2007), 127147. MR 2296069 (2008a:55010)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
55Q99,
55Q52
Retrieve articles in all journals
with MSC (2000):
55Q99,
55Q52
Additional Information
Liu Xiugui
Affiliation:
School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Wang Xiangjun
Affiliation:
School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Yuan Zihong
Affiliation:
School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email:
yuanzhchina@gmail.com
DOI:
http://dx.doi.org/10.1090/S0002993909101387
Keywords:
Stable homotopy,
AdamsNovikov spectral sequence,
chromatic spectral sequence
Received by editor(s):
August 1, 2008
Received by editor(s) in revised form:
July 22, 2009
Published electronically:
October 28, 2009
Additional Notes:
The authors were partially supported by NSFC grant No. 10771105.
Communicated by:
Paul Goerss
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
