The Zero Slice of Quaternionic Real Bordism

TL;DR

This paper computes the zero slice of a quaternionic MU spectrum constructed via the Hill-Hopkins-Ravenel norm, advancing the understanding of its slice spectral sequence and homotopy Mackey functors.

Contribution

It introduces the computation of the zero slice and a subring of homotopy Mackey functors for the quaternionic MU spectrum using the Hill-Hopkins-Ravenel norm.

Findings

Computed the zero slice of the $Q_8$-spectrum $N_{C_2}^{Q_8} ext{MU} ext{R}$.

Identified a bigraded subring of the RO($Q_8$)-graded homotopy Mackey functors.

Progressed towards the full computation of the slice spectral sequence for this spectrum.

Abstract

Using the Hill-Hopkins-Ravenel norm, one can produce a $Q_8$-spectrum $N_{C_2}^{Q_8} \text{MU}\mathbb{R}$, where $Q_8$ is the quaternion group. Working towards a computation of the slice spectral sequence for $N_{C_2}^{Q_8} \text{MU}\mathbb{R}$, we compute the zero slice of $N_{C_2}^{Q_8} \text{MU}\mathbb{R}$ and a bigraded subring of the $\text{RO}(Q_8)$-graded homotopy Mackey functors of this slice.