Brown-Peterson cohomology from Morava $K$-theory
This is the "final" version of this paper. It is significantly
changed from the version which has been on the archive.
Douglas C. Ravenel
University of Rochester
Rochester, New York 14627
drav@troi.cc.rochester.edu}
W. Stephen Wilson
Johns Hopkins University
Baltimore, Maryland 21218
wsw@math.jhu.edu
Nobuaki Yagita
Ibaraki University
Mito, Ibaraki, Japan
yagita@mito.ipc.ibaraki.ac.jp
We give some structure to the Brown-Peterson cohomology (or its
$p$-completion) of a wide class of spaces. The class of spaces
are those with Morava K-theory even dimensional. We can say
that the Brown-Peterson cohomology is even dimensional
(concentrated in even degrees) and is flat as a $BP^*$-module
for the category of finitely presented $BP^*(BP)$-modules. At
first glance this would seem to be a very restricted class of
spaces but the world abounds with naturally occurring examples:
Eilenberg-MacLane spaces, loops of finite Postnikov systems,
classifying spaces of most finite groups whose Morava K-theory
is known (including the symmetric groups), $QS^{2n}$, $BO(n)$,
$MO(n)$, $BO$, $\ImJ$, etc. We finish with an explicit
algebraic construction of the Brown-Peterson cohomology of a
product of Eilenberg-MacLane spaces and a general K\"unneth
isomorphism applicable to our situation.