Some remarks on Real and algebraic cobordism, by Po Hu and Igor Kriz

In this paper, we discuss certain conjectures and theorems about algebraic cobordism and algebraic Morava K(n)-theories. Concretely, we describe a theory of algebraically oriented spectra as an analogue of complex and Real-oriented spectra and some conjectures about MGL suggested by this point of view. We then apply this philosophy to the idea of splitting algebraicaaly-oriented cohomology of Pfister quadrics in a way that would generalized Rost's splitting theorem for motivic cohomology. For example, we construct a twisted Morava K-theory K(n)^\perp which generalizes symplectic K-theory, and modulo some conjectures about the coefficients of MGL, prove that algebraic Morava K(n)-theories of certain quadrics considered by Pfister split as sums of K(n) and K(n)^\perp. We also construct an algebraic spectrum whose motive is the Rost motive (for \ell=2), and consider an algebraic analogue of the Hopf invariant one problem.


Po Hu <pohu@math.uchicago.edu>
Igor Kriz <ikriz@math.lsa.umich.edu>