Brown-Peterson cohomology from Morava K-theory
Douglas C. Ravenel W. Stephen Wilson and Nobuaki Yagita
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 all 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.