Motivic Toda brackets II
Xiaowen Dong
TL;DR
The paper addresses constructing unstable motivic Toda brackets over $\mathrm{Spec}\mathbb{Z}$ using the first two Hopf maps $\eta$ and $\nu$ in the unstable regime. It employs join/cone techniques, explicit diagram-chasing, and a geometric realization functor to prove unstable null-Hopf relations, enabling the definition of Toda brackets $\{\eta_{3+(2)}, \nu_{3+(3)}, \eta_{4+(5)}\}$, $\{\nu_{3+(3)}, \eta_{4+(5)}, \nu_{4+(6)}\}$, and $\{\eta_{3+(2)}, (1-\epsilon)_{3+(3)}, \nu_{3+(3)}\circ \nu_{4+(5)}\}$, with key relations such as $1_{1+(2)}+\eta_{1+(2)}\circ \Delta_{1+(3)}=-\epsilon_{1+(2)}$ and $\eta_{3+(2)}\circ \nu_{3+(3)}=0$, $\nu_{3+(3)}\circ \eta_{4+(5)}=0$. The complex realization of these brackets maps to nontrivial topological Toda brackets, confirming their nontriviality and connecting unstable motivic phenomena to classical topology. Together, these results identify the smallest nontrivial unstable bidegrees and expand the toolkit for higher-order unstable operations in motivic homotopy theory.
Abstract
We construct more non-trivial examples for Toda brackets in unstable motivic homotopy theory via the first and second motivic Hopf maps.