On the fate of $\eta^3$ in higher analogues of Real bordism

Abstract

We show that the cube of the Hopf map $\eta$ maps to zero under the Hurewicz map for all fixed points of all norms to cyclic $2$-groups of the Landweber-Araki Real bordism spectrum. Using that the slice spectral sequence is a spectral sequence of Mackey functors, we compute the relevant portion of the homotopy groups of these fixed points, showing that multiplication by $4$ annihilates $\pi_{3}$.