Multiplicative Equivariant Thom Spectra & Structured Real Orientations

Abstract

For strongly even $\mathbb{E}_{\infty}^{C_2}$-rings $E$ we show that any homotopy ring map $\mathrm{MU} \to E^e$ lifts to an $\mathbb{E}_{\rho}$-map $\mathrm{MU}_{\mathbb{R}} \to E$. This refines the Hahn-Shi Real orientations of Lubin-Tate theories $E_n$, the Hirzebruch level-$n$ orientations of $\mathrm{tmf}_1(n)$, and Quillen's idempotent to $\mathbb{E}_\rho$-maps. It allows us to provide the first structured version of $\mathrm{BP}_{\mathbb{R}}$ - we show that it admits an $\mathbb{E}_{\rho}$-algebra structure. Furthermore, we extend these results to larger groups. In particular, for a finite group $C_2 \leq G$ the Hahn-Shi orientation $N_{C_2}^G \mathrm{MU}_{\mathbb{R}} \to E_n$ refines to a $\operatorname{Coind}_{C_2}^G \mathbb{E}_{\rho}$-map, and $N^G_{C_2}\mathrm{BP}_{\mathbb{R}}$ admits a $\operatorname{Coind}_{C_2}^G \mathbb{E}_{\rho}$-algebra structure. Essential to this program is a robust theory of multiplicative equivariant Thom spectra, which we develop using parametrized higher algebra and fibrous patterns - particularly, we provide an equivariant version of Antol\'in-Camarena--Barthel's universal property for multiplicative Thom spectra and use this to deduce a multiplicative equivariant Thom isomorphism. We provide a number of categorical results of independent interest, most notably a distributive monoidal structure on parametrized left module categories.