The complex oriented cohomology of extended powers

John Robert Hunton

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 2, page 517-534
  • ISSN: 0373-0956

Abstract

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We examine the behaviour of a complex oriented cohomology theory G * ( - ) on D p ( X ) , the C p -extended power of a space X , seeking a description of G * ( D p ( X ) ) in terms of the cohomology G * ( X ) . We give descriptions for the particular cases of Morava K -theory K ( n ) for any space X and for complex cobordism M U , the Brown-Peterson theories BP and any Landweber exact theory for a wide class of spaces.

How to cite

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Hunton, John Robert. "The complex oriented cohomology of extended powers." Annales de l'institut Fourier 48.2 (1998): 517-534. <http://eudml.org/doc/75291>.

@article{Hunton1998,
abstract = {We examine the behaviour of a complex oriented cohomology theory $G^*(-)$ on $D_p(X)$, the $C_p$-extended power of a space $X$, seeking a description of $G^*(D_p(X))$ in terms of the cohomology $G^*(X)$. We give descriptions for the particular cases of Morava $K$-theory $K(n)$ for any space $X$ and for complex cobordism $MU$, the Brown-Peterson theories BP and any Landweber exact theory for a wide class of spaces.},
author = {Hunton, John Robert},
journal = {Annales de l'institut Fourier},
keywords = {extended power of a space; complex oriented cohomology; Morava K-theory; Landweber exact cohomology theories; complex cobordism},
language = {eng},
number = {2},
pages = {517-534},
publisher = {Association des Annales de l'Institut Fourier},
title = {The complex oriented cohomology of extended powers},
url = {http://eudml.org/doc/75291},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Hunton, John Robert
TI - The complex oriented cohomology of extended powers
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 2
SP - 517
EP - 534
AB - We examine the behaviour of a complex oriented cohomology theory $G^*(-)$ on $D_p(X)$, the $C_p$-extended power of a space $X$, seeking a description of $G^*(D_p(X))$ in terms of the cohomology $G^*(X)$. We give descriptions for the particular cases of Morava $K$-theory $K(n)$ for any space $X$ and for complex cobordism $MU$, the Brown-Peterson theories BP and any Landweber exact theory for a wide class of spaces.
LA - eng
KW - extended power of a space; complex oriented cohomology; Morava K-theory; Landweber exact cohomology theories; complex cobordism
UR - http://eudml.org/doc/75291
ER -

References

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  1. [1] A.J. BAKER and J.R. HUNTON, Continuous Morava K-theory and the geometry of the In-adic tower, Math. Scand., 75 (1994), 67-81. Zbl0828.55003MR96a:55009
  2. [2] A.J. BAKER and U. WÜRGLER, Bockstein operations in Morava K-theories, Forum Math., 3 (1991), 543-560. Zbl0751.55002MR92i:55009
  3. [3] R. BRUNER, J.P. MAY, J.E. MCCLURE and M. STEINBERGER, H∞ ring spectra and their applications, Springer Lecture Notes in Math., vol. 1176 (1986). Zbl0585.55016MR88e:55001
  4. [4] A.D. ELMENDORF, I. KRIZ, M.A. MANDELL, J.P. MAY, Modern foundations for stable homotopy theory, Handbook of Algebraic Topology, editor I. M. James, (1995) Elsevier North-Holland. Zbl0865.55007MR97d:55016
  5. [5] M.J. HOPKINS and J.R. HUNTON, On the structure of spaces representing a Landweber exact cohomology theory, Topology, 34 (1995), 29-36. Zbl0862.55005MR95k:55009
  6. [6] M. HOVEY, Bousfield Localisation functors and Hopkins' chromatic splitting conjecture, Proceedings of the Čech Centennial Homotopy conference, June 1993, American Mathematical Society Contemporary Mathematics Series, editors Mila Cenkl and Haynes Miller, 181 (1995), 225-250. Zbl0830.55004
  7. [7] M. HOVEY and H. SADOFSKI, Invertible spectra in the E(n) local stable homotopy category, to appear, Journal of the London Mathematical Society. Zbl0947.55013
  8. [8] M. HOVEY and N. STRICKLAND, Morava K-theories and localisation, preprint. Zbl0929.55010
  9. [9] J.R. HUNTON, The Morava K-theory of wreath products, Math. Proc. Camb. Phil. Soc., 107 (1990), 309-318. Zbl0705.55009MR91a:55004
  10. [10] J.R. HUNTON, Detruncating Morava K-theory, Proc. Adams Memorial Symposium, LMS Lecture notes series, C.U.P., 176 (1992) 35-43. Zbl0751.55003MR94k:55010
  11. [11] J.R. HUNTON and P.R. TURNER, An exactness theorem for the homology of representing spaces, preprint. Zbl0929.55007
  12. [12] T. KASHIWABARA, On Brown-Peterson cohomology of QX, preprint. Zbl0985.55006
  13. [13] P.S. LANDWEBER, Homological properties of comodules over MU*(MU) and BP*(BP), Amer. J. Math., 98 (1976), 591-610. Zbl0355.55007MR54 #11311
  14. [14] D. LAZARD, Autour de la platitude, Bull. Soc. Math. France, 97 (1969), 81-128. Zbl0174.33301MR40 #7310
  15. [15] I.J. LEARY, On the integral cohomology of wreath products, to appear, J. Algebra. Zbl0893.55009
  16. [16] L.G. LEWIS, Jr., J.P. MAY and M. STEINBERGER, Equivariant stable homotopy theory, Springer Lecture Notes in Math., vol. 1213 (1986). Zbl0611.55001MR88e:55002
  17. [17] J.E. MCCLURE and V.P. SNAITH, On the K-theory of the extended power construction, Math. Proc. Camb. Phil. Soc., 92 (1982), 263-274. Zbl0508.55021MR83j:55004
  18. [18] J. MILNOR, The Steenrod algebra and its dual, Ann. Math., 67 (1958), 150-171. Zbl0080.38003MR20 #6092
  19. [19] M. NAKAOKA, Homology of the infinite symmetric group, Ann. Math., 73 (1961), 229-257. Zbl0099.25301MR24 #A1721
  20. [20] D.C. RAVENEL and W.S. WILSON, The Hopf ring for complex cobordism, Journal of Pure and Applied Algebra, 9 (1977), 241-280. Zbl0373.57020MR56 #6644
  21. [21] D.C. RAVENEL and W.S. WILSON, The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture, Amer. J. Math., 102 (1980), 691-748. Zbl0466.55007MR81i:55005
  22. [22] D.C. RAVENEL, W.S. WILSON and N. YAGITA, Brown-Peterson cohomology from Morava K-theory, to appear, Journal of K-theory. Zbl0912.55002
  23. [23] V.P. SNAITH, A stable decomposition of ΩnΣnX, J. London Math. Soc., 7 (1974), 577-583. Zbl0275.55019MR49 #3918
  24. [24] D. TAMAKI, Ph. D. thesis, University of Rochester. 
  25. [25] U. WÜRGLER, On products in a family of cohomology theories associated to the invariant prime ideals of π*(BP), Comment. Math. Helv., 52 (1977), 457-481. Zbl0379.55002MR57 #17624
  26. [26] N. YAGITA, On the Steenrod algebra of Morava K-theory, J. London Math. Soc., (2) 22 (1980), 423-438. Zbl0453.55013MR82f:55027

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