Multiplicative Equivariant Thom Spectra & Structured Real Orientations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913025659764736 |
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| author | Quinn, Ryan Zhu, Qi |
| author_facet | Quinn, Ryan Zhu, Qi |
| contents | 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 $\mathbb{E}_ρ$-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 $\mathbb{E}_ρ$-maps. It allows us to provide the first structured version of $\mathrm{BP}_{\mathbb{R}}$ - we show that it admits an $\mathbb{E}_ρ$-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 $\operatorname{Coind}_{C_2}^G \mathbb{E}_ρ$-map, and $N^G_{C_2}\mathrm{BP}_{\mathbb{R}}$ admits a $\operatorname{Coind}_{C_2}^G \mathbb{E}_ρ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicative Equivariant Thom Spectra & Structured Real Orientations Quinn, Ryan Zhu, Qi Algebraic Topology Category Theory 55P91, 18N70, 55P43 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 $\mathbb{E}_ρ$-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 $\mathbb{E}_ρ$-maps. It allows us to provide the first structured version of $\mathrm{BP}_{\mathbb{R}}$ - we show that it admits an $\mathbb{E}_ρ$-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 $\operatorname{Coind}_{C_2}^G \mathbb{E}_ρ$-map, and $N^G_{C_2}\mathrm{BP}_{\mathbb{R}}$ admits a $\operatorname{Coind}_{C_2}^G \mathbb{E}_ρ$-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. |
| title | Multiplicative Equivariant Thom Spectra & Structured Real Orientations |
| topic | Algebraic Topology Category Theory 55P91, 18N70, 55P43 |
| url | https://arxiv.org/abs/2512.15573 |