Real Global Group Laws and Hu-Kriz Maps

TL;DR

This paper integrates global group laws with Hu-Kriz maps using $C_2$-global spectra, establishing new isomorphisms and proposing an evenness conjecture in equivariant stable homotopy theory.

Contribution

It introduces a unified framework for global and equivariant spectra, generalizes $MU_G$ and $M\mathbb{R}$, and proves split surjections and conjectures about isomorphisms.

Findings

Restriction maps are split surjections in certain semi-direct product cases.

Proposed an evenness conjecture implying isomorphisms.

Defined new notions of Real orientations and global group laws.

Abstract

Recently, Hausmann defined global group laws and used them to prove that $MU^G_*$ is the $G$-equivariant Lazard ring, for $G$ a compact abelian Lie group. On the other hand, Hu and Kriz showed that the restriction map induces an isomorphism $M \mathbb{R}^{C_2}_{\rho *} \cong MU_{2*}$. In this paper, we blend these stories. We utilize the $C_2$-global spectrum $\mathbf{MR}$ defined by Schwede in an unpublished note, which gives rise to a genuine $G$-spectrum $M \mathbb{R}_\eta$ for each augmented compact Lie groups $\eta: G\to C_2$, simultaneously generalizing $MU_G$ and $M \mathbb{R}$. In the case of semi-direct product augmentations $G \rtimes C_2\to C_2$ with $G$ compact abelian Lie and $C_2$ acting by inversion, we show that the restriction along the inclusion $G \subset G \rtimes C_2$ is a split surjection $M \mathbb{R}^{G \rtimes C_2}_{\rho *} \rightarrow MU^{G}_{2*}$. Additionally, we propose an evenness conjecture, which implies that this map is an isomorphism. Along the way, we define Real $\eta$-orientations, Real global orientations, and corresponding notions of equivariant and global group laws.