Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

From $K(n+1)_*(X)$ to $K(n)_*(X)$


Author: Norihiko Minami
Journal: Proc. Amer. Math. Soc. 130 (2002), 1557-1562
MSC (2000): Primary 55N15, 55N20, 55N22; Secondary 55Q51, 55R35
DOI: https://doi.org/10.1090/S0002-9939-01-06374-2
Published electronically: October 12, 2001
MathSciNet review: 1879983
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $X$ be a space of finite type. Set $q=2(p-1)$ as usual, and define the mod $q$ support of $K(n)^*(X)$ by $ S(X,K(n)) = \{ m \in { \mathbb Z}/q{ \mathbb Z}\mid \bigoplus_{d \equiv m \operatorname{mod}q} K(n)^d \neq 0 \} $ for $n>0.$ Call $K(n)^*(X)$sparse if there is no $m \in { \mathbb Z}/q{ \mathbb Z}$ with $ m, m+1 \in S(X,K(n)). $

Then we show the relation $ S(X,K(n)) \subseteqq S(X,K(n+1)) $ for any finite type space $X$ with $K(n+1)^*(X)$ being sparse.

As a special case, we have $ K(n+1)^{odd}(X) = 0 \Longrightarrow K(n)^{odd}(X) = 0, $ and the main theorem of Ravenel, Wilson and Yagita is also generalized in terms of the mod $q$ support.


References [Enhancements On Off] (What's this?)

  • 1. J. M. Boardman, Stable operations in generalized cohomology, In I. M. James, editor, The Handbook of Algebraic Topology, chapter 14, 585-686. Elsevier, 1995. MR 97b:55021
  • 2. J. M. Boardman, D. C. Johnson and W. S. Wilson, Unstable operations in generalized cohomology, In I. M. James, editor, The Handbook of Algebraic Topology, chapter 15, 687-828. Elsevier, 1995. MR 97b:55022
  • 3. A. K. Bousfield, On $K(n)$-equivalence of spaces, Contemp. Math. 239 (1999), 85-89. CMP 2000:03
  • 4. M. Hovey, Cohomological Bousfield Classes, J. Pure Appl. Algebra 103 (1995), 45-59. MR 96g:55008
  • 5. M. Hovey, Bousfield localization functors and Hopkins' chromatic splitting conjecture, Contemp. Math. 181 (1995), 225-250. MR 96m:55010
  • 6. I. Kriz, Morava $K$-theory of classifying spaces: some calculations, Topology 36 (1997), 1247-1273. MR 99a:55016
  • 7. I. Kriz and K. P. Lee, Odd-degree elements in the Morava $K(n)$-cohomology of finite groups, Topology Appl. 103 (2000), 229-241. MR 2001f:55008
  • 8. N. Minami, On the chromatic tower, in preparation.
  • 9. D. C. Ravenel, W. S. Wilson and N. Yagita, Brown-Peterson Cohomology from Morava $K$-Theory, K-Theory 15 (1998), 147-199. MR 2000d:55012
  • 10. W. S. Wilson, $K(n+1)$ equivalence implies $K(n)$ equivalence, Contemp. Math. 239 (1999), 375-376. MR 2000i:55021
  • 11. N. Yagita, The exact functor theorem for $BP_*/I_n$-theory, Proc. Japan Acad. 52 (1976), 1-3. MR 52:15432
  • 12. Z. Yosimura, Projective dimension of Brown-Peterson homology with modulo $(p, v_1, \ldots, v_{n-1})$coefficients, Osaka J. Math. 13 (1976), 289-309. MR 54:3686

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 55N15, 55N20, 55N22, 55Q51, 55R35

Retrieve articles in all journals with MSC (2000): 55N15, 55N20, 55N22, 55Q51, 55R35


Additional Information

Norihiko Minami
Affiliation: Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa-ku, Nagoya 466-8555, Japan
Email: norihiko@math.kyy.nitech.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-01-06374-2
Keywords: Morava $K$-theory, unstable homotopy theory, classifying space
Received by editor(s): November 20, 2000
Published electronically: October 12, 2001
Additional Notes: This research was partially supported by Grant-in-Aid for Scientific Research No. 11640072, Japan Society for the Promotion of Science
Communicated by: Ralph Cohen
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society