Morava Hopf algebras and spaces K(n) equivalent to finite Postnikov systems

We have three somewhat independent sets of results. Our first results are a mixed blessing. We show that Morava K-theories don’t see k-invariants for homotopy commutative H-spaces which are finite Postnikov systems, i.e. for those with only a finite number of homotopy groups. Since k-invariants are what holds the space together, this suggests that Morava K-theories will not be of much use around such spaces. On the other hand, this gives us the Morava K-theory of a wide class of spaces which is bound to be useful. In particular, this work allows the recent work in [RWY] to be applied to compute the Brown-Peterson cohomology of all such spaces. Their Brown-Peterson cohomology turns out to be all in even degrees (as is their Morava Ktheory) and flat as a BP ∗ module for the category of finitely presented ∗Partially supported by the National Science Foundation †Partially supported by the National Science Foundation