Brown–Peterson Cohomology from Morava K-Theory

Abstract

Abstract. 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), 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-MacLane spaces and a general Künneth isomorphism applicable to our situation. 1.