The Morava K-theory and Brown-Peterson cohomology
of spaces related to BP
Takuji Kashiwabara
Institut Fourier, Universit\'{e} de Grenoble I, U.M.R. au C.N.R.S.,
B. P. 74, 38402 Saint-Martin-d'H\`{e}res
CEDEX France
Takuji.Kashiwabara@ujf-grenoble.fr
W. Stephen Wilson
Department of Mathematics
Johns Hopkins University
Baltimore, Maryland 21218
wsw@math.jhu.edu
This is the "final" version of the paper.
We calculate the Morava K-theory of the spaces in the Omega
spectra for BP. They fit into an exotic array of short
and long exact sequences of Hopf algebras. We apply this to
calculate the p-adically completed Brown-Peterson cohomology,
as well as all of the intermediary cohomology theories, E, of
these spaces. We give two descriptions of the answer, both
of which turn out to be surprisingly nice. One part of our
first description is just the image in the E cohomology of
the corresponding space in the Omega spectrum for BP, which
is as big as it could possibly be and which we show how to
calculate. The other part is just the E cohomology of
several copies of Eilenberg-MacLane spaces, something which
is already known. Our second description is inductive and
gives us a new way of looking at the Brown-Peterson
cohomology of Eilenberg-MacLane spaces. The Brown-Comenetz
dual of BP shows up in our calculations and so we take up
the study of this spectrum as well. It was already known
that the Morava K-theory of the spaces in the Omega spectrum
for the Brown-Comenetz dual of BP made it look like a
product of Eilenberg-MacLane spaces and we find, somewhat to
our surprise, that the same is true for the BP cohomology.
In order to state our answers we set up the foundations for
the category of completed Hopf algebras.