2025/12/17 by Quinn, Ryan, Zhu, Qi
Mathematics · #18N70 #55P43 #55P91 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.48550/arxiv.2512.15573
openalex publication_date 2025/12/17 · openalex created_date 2025/12/19 · openalex updated_date 2026/07/28
For strongly even 𝔼∞C2-rings E we show that any homotopy ring map MU → Ee lifts to an 𝔼ρ-map MUℝ → E. This refines the Hahn-Shi Real orientations of Lubin-Tate theories En, the Hirzebruch level-n orientations of tmf1(n), and Quillen's idempotent to 𝔼ρ-maps. It allows us to provide the first structured version of BPℝ - we show that it admits an 𝔼ρ-algebra structure. Furthermore, we extend these results to larger groups. In particular, for a finite group C2 ≤ G the Hahn-Shi orientation NC2G MUℝ → En refines to a CoindC2G 𝔼ρ-map, and NGC2BPℝ admits a CoindC2G 𝔼ρ-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.