The Morava K-theory of BO(q) and MO(q)

TL;DR

This paper provides a simplified proof that Morava K-theories of BO(q) and MO(q) are concentrated in even degrees, and explicitly computes their homology, connecting geometric and algebraic perspectives.

Contribution

It offers an easier proof of even-degree Morava K-theories for BO(q) and MO(q), and explicitly describes their homology, linking geometric and algebraic approaches.

Findings

Morava K-theories of BO(q) and MO(q) are in even degrees

Explicit descriptions of K(n)_*(BO(q)) and K(n)_*(MO(q)) are provided

Connection established between geometric homology and algebraic Landweber flatness

Abstract

We give an easy proof that the Morava K-theories for BO(q) and MO(q) are in even degrees. Although this is a known result, it had followed from a difficult proof that BP^*(BO(q)) was Landweber flat. Landweber flatness follows from the even Morava K-theory. We go further and compute an explicit description of K(n)_*(BO(q)) and K(n)_*(MO(q)) and reconcile it with the purely algebraic construct from Landweber flatness.