Transchromatic extensions in motivic and Real bordism

Abstract

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

References in corpus (3)

• Universal Toda brackets of ring spectra
• Models of Lubin-Tate spectra via Real bordism theory
• Algebraic cobordism of number fields

Cited by in corpus (1)

• Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf{C}$