The complex oriented cohomology of extended powers
Annales de l'Institut Fourier, Volume 48 (1998) no. 2, pp. 517-534.

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.

Nous étudions le comportement d’une théorie à orientation complexe G * (-) sur un espace du type D p (X), la puissance C p -étendue d’un espace X, à la recherche d’une description de G * (D p (X)) en fonction de G * (X). Nous donnons une telle description dans le cas particulier des K-théories de Morava K(n) (pour X espace quelconque) et dans le cas du cobordisme complexe MU, de la théorie de Brown-Peterson BP ou de n’importe quelle théorie Landweber-exacte, pour X décrivant une vaste classe d’espaces.

@article{AIF_1998__48_2_517_0,
     author = {Hunton, John Robert},
     title = {The complex oriented cohomology of extended powers},
     journal = {Annales de l'Institut Fourier},
     pages = {517--534},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {2},
     year = {1998},
     doi = {10.5802/aif.1627},
     zbl = {0899.55019},
     mrnumber = {99c:55017},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1627/}
}
TY  - JOUR
AU  - Hunton, John Robert
TI  - The complex oriented cohomology of extended powers
JO  - Annales de l'Institut Fourier
PY  - 1998
DA  - 1998///
SP  - 517
EP  - 534
VL  - 48
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1627/
UR  - https://zbmath.org/?q=an%3A0899.55019
UR  - https://www.ams.org/mathscinet-getitem?mr=99c:55017
UR  - https://doi.org/10.5802/aif.1627
DO  - 10.5802/aif.1627
LA  - en
ID  - AIF_1998__48_2_517_0
ER  - 
%0 Journal Article
%A Hunton, John Robert
%T The complex oriented cohomology of extended powers
%J Annales de l'Institut Fourier
%D 1998
%P 517-534
%V 48
%N 2
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1627
%R 10.5802/aif.1627
%G en
%F AIF_1998__48_2_517_0
Hunton, John Robert. The complex oriented cohomology of extended powers. Annales de l'Institut Fourier, Volume 48 (1998) no. 2, pp. 517-534. doi : 10.5802/aif.1627. http://archive.numdam.org/articles/10.5802/aif.1627/

[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. | EuDML | MR | Zbl

[2] A.J. Baker and U. Würgler, Bockstein operations in Morava K-theories, Forum Math., 3 (1991), 543-560. | EuDML | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[5] M.J. Hopkins and J.R. Hunton, On the structure of spaces representing a Landweber exact cohomology theory, Topology, 34 (1995), 29-36. | MR | Zbl

[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. | MR | Zbl

[7] M. Hovey and H. Sadofski, Invertible spectra in the E(n) local stable homotopy category, to appear, Journal of the London Mathematical Society. | MR | Zbl

[8] M. Hovey and N. Strickland, Morava K-theories and localisation, preprint. | MR | Zbl

[9] J.R. Hunton, The Morava K-theory of wreath products, Math. Proc. Camb. Phil. Soc., 107 (1990), 309-318. | MR | Zbl

[10] J.R. Hunton, Detruncating Morava K-theory, Proc. Adams Memorial Symposium, LMS Lecture notes series, C.U.P., 176 (1992) 35-43. | MR | Zbl

[11] J.R. Hunton and P.R. Turner, An exactness theorem for the homology of representing spaces, preprint.

[12] T. Kashiwabara, On Brown-Peterson cohomology of QX, preprint. | Zbl

[13] P.S. Landweber, Homological properties of comodules over MU*(MU) and BP*(BP), Amer. J. Math., 98 (1976), 591-610. | MR | Zbl

[14] D. Lazard, Autour de la platitude, Bull. Soc. Math. France, 97 (1969), 81-128. | Numdam | MR | Zbl

[15] I.J. Leary, On the integral cohomology of wreath products, to appear, J. Algebra. | Zbl

[16] L.G. Lewis, Jr., J.P. May and M. Steinberger, Equivariant stable homotopy theory, Springer Lecture Notes in Math., vol. 1213 (1986). | MR | Zbl

[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. | MR | Zbl

[18] J. Milnor, The Steenrod algebra and its dual, Ann. Math., 67 (1958), 150-171. | MR | Zbl

[19] M. Nakaoka, Homology of the infinite symmetric group, Ann. Math., 73 (1961), 229-257. | MR | Zbl

[20] D.C. Ravenel and W.S. Wilson, The Hopf ring for complex cobordism, Journal of Pure and Applied Algebra, 9 (1977), 241-280. | MR | Zbl

[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. | MR | Zbl

[22] D.C. Ravenel, W.S. Wilson and N. Yagita, Brown-Peterson cohomology from Morava K-theory, to appear, Journal of K-theory. | Zbl

[23] V.P. Snaith, A stable decomposition of ΩnΣnX, J. London Math. Soc., 7 (1974), 577-583. | MR | Zbl

[24] D. Tamaki, Ph. D. thesis, University of Rochester.

[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. | MR | Zbl

[26] N. Yagita, On the Steenrod algebra of Morava K-theory, J. London Math. Soc., (2) 22 (1980), 423-438. | MR | Zbl

Cited by Sources: