Topological Morava K-theory of Lie groups (structure unknown)

Determine the topological Morava K-theory K(n)^{top}(K) of simple compact Lie groups, providing a complete description of its multiplication and co-multiplication structures; in particular, ascertain the full ring and Hopf algebra structures of K(n)^{top}(SO(m)) and K(n)^{top}(Spin(m)).

Background

The paper surveys the state of knowledge on topological Morava K-theory for Lie groups, noting that while some additive information is known, multiplicative and coalgebraic structures are largely unresolved.

For orthogonal and spinor groups, only the additive structure had been computed previously, with multiplication and co-multiplication remaining an open area; partial results exist but a general solution is lacking.