Transchromatic extensions in motivic and Real bordism

TL;DR

This paper explores chromatic height shifts in motivic and Real bordism spectra via Toda brackets, revealing exotic multiplications and non-trivial elements in their homotopy groups.

Contribution

It introduces red-shifting Toda brackets in motivic and Real bordism spectra, leading to new exotic multiplications in their homotopy modules.

Findings

Identification of red-shifting Toda brackets in $MGL$ and $MU_{\mathbb R}$

Discovery of exotic multiplications including non-trivial 2-multiplications

Implications for the structure of motivic and Real Morava K-theories

Abstract

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum $MGL$ and of the Real bordism spectrum $MU_{\mathbb R}$. These brackets are "red-shifting" in the sense that while the terms in the bracket will be of some chromatic height $n$, 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 $\pi_{\star}MU_\mathbb R$-module structure on the Real Morava $K$-theories.