Brown–Peterson Cohomology from Morava K-Theory

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-Mac Lane spaces, loops of finite Postnikov systems, classifying spaces of most finite groups whose Morava K-theory is known (including the symmetric groups), QS²ⁿ, BO(n), MO(n), BO, Im J, etc. We finish with an explicit algebraic construction of the Brown–Peterson cohomology of a product of Eilenberg–Mac Lane spaces and a general Künneth isomorphism applicable to our situation.