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

TL;DR

This paper investigates the behavior of the Hopf map $ ext{eta}$ in higher analogues of Real bordism, revealing that its cube maps to zero and that multiplication by 4 annihilates certain homotopy groups.

Contribution

It demonstrates that the cube of the Hopf map $ ext{eta}$ maps to zero in all fixed points of Landweber-Araki Real bordism spectra and computes related homotopy groups using Mackey functors.

Findings

The cube of $ ext{eta}$ maps to zero under the Hurewicz map.

Multiplication by 4 annihilates $ ext{pi}_3$ of fixed points.

Computed the relevant homotopy groups using the slice spectral sequence.

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 $\pi_{3}$.