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 ${\mathbb{E}}_{\ddot{I}}$-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 ${\mathbb{E}}_{\ddot{I}}$-maps. It allows us to provide the first structured version of ${\mathrm{BP}}_{\mathbb{R}}$ - we show that it admits an ${\mathbb{E}}_{\ddot{I}}$-algebra structure. Furthermore, we extend these results to larger groups. In particular, for a finite group $C_2\le G$ the Hahn-Shi orientation $N_{C_2}^G{\mathrm{MU}}_{\mathbb{R}}\to E_n$ refines to a ${\mathrm{Coind}}_{C_2}^G{\mathbb{E}}_{\ddot{I}}$-map, and $N_{C_2}^G{\mathrm{BP}}_{\mathbb{R}}$ admits a ${\mathrm{Coind}}_{C_2}^G{\mathbb{E}}_{\ddot{I}}$-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 Antolín-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.

92 pages, various improvements, comments still very welcome!