Transchromatic extensions in motivic bordism

Abstract

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum M G L MGL and of the Real bordism spectrum M U R MU_{\mathbb R} . These brackets are “red-shifting” in the sense that while the terms in the bracket will be of some chromatic height n n , the bracket itself will be of chromatic height ( n + 1 ) (n+1) . Using these, we deduce a family of exotic multiplications in the π ( ∗ , ∗ ) M G L \pi _{(\ast ,\ast )}MGL -module structure of the motivic Morava K K -theories, including non-trivial multiplications by 2 2 . These in turn imply the analogous family of exotic multiplications in the π ⋆ M U R \pi _{\star }MU_\mathbb R -module structure on the Real Morava K K -theories.