Real Global Group Laws and Hu-Kriz Maps

TL;DR

This work fuses Hausmann’s global group laws with Hu–Kriz’s equivariant isomorphism through Real global homotopy theory. It introduces the Real spectrum $\mathbf{MR}$ and its augments $M\mathbb{R}_\eta$ to relate $MU_G$ and $M\mathbb{R}$ across augmented Lie groups, yielding Real $G$-equivariant and Real global orientations that induce Real (equivariant) formal group laws and a Real global group law. A central result shows that for semi-direct product augmentations $G\rtimes C_2\to C_2$ with compact abelian $G$, the restriction map $\pi^{\hat G}_{\rho*}(M\mathbb{R}_{\hat G})\to\pi^G_{2*}(MU_G)$ is a split surjection, with proposed Evenness and Regularity conjectures ensuring it is an isomorphism. These constructions yield a robust framework to compare global and equivariant chromatic structures and pave the way for automatic transfer of universal orientation data from $MU_G$ to Real global spectra. The results have potential to unify perspectives on global group laws and equivariant cobordism in the presence of Real structures.

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 $M \mathbb{R}^{C_2}_{ρ*} \cong MU_{2*}$. 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 We utilize the $C_2$-global spectrum $\mathbf{MR}$ defined by Schwede in an unpublished note, which gives rise to a genuine $G$-spectrum $M \mathbb{R}_η$ for each augmented compact Lie groups $η: G\to C_2$, simultaneously generalizing $MU_G$ and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 $M \mathbb{R}^{G \rtimes C_2}_{ρ*} \rightarrow MU^{G}_{2*}$. Additionally, we propose an evenness conjecture, which implies that this map is an isomorphism. Along the way, we define Real $η$-orientations, Real global orientations, and corresponding notions of equivariant and global group laws.

Theorems & Definitions (108)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3: Evenness Conjecture, Conjecture \ref{['conj:MR-eta-is-even']}
• Conjecture 1.4: Regularity conjecture, Conjecture \ref{['conj:R-is-regular']}
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Definition 2.2
• Proposition 2.3: Sch14