Title:

The Morava K-theory and Brown-Peterson cohomology of spaces related to BP

Authors:

Takuji Kashiwabara 
W. Stephen Wilson 

Addresses:

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

Department of Mathematics 
Johns Hopkins University 
Baltimore, Maryland 21218 
and
Department of Mathematics 
Kyoto University 
Kyoto 606-8502 Japan

Emails:

Takuji.Kashiwabara@ujf-grenoble.fr
wsw@math.jhu.edu

Abstract:

We calculate the Morava K-theory of the spaces in the Omega
spectra for BP<q>.  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<q>
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<q> 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.