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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.06222 |
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| _version_ | 1866918152851423232 |
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| author | Nocera, Guglielmo Porzio, Morena |
| author_facet | Nocera, Guglielmo Porzio, Morena |
| contents | Let $G$ be a complex reductive group. A folklore result asserts the existence of an $\mathbb{E}_2$-algebra structure on the Ran Grassmannian of $G$ over $\mathbb{A}^1_{\mathbb{C}}$, seen as a topological space with the complex-analytic topology. The aim of this paper is to prove this theorem, by establishing a homotopy invariance result: namely, an inclusion of open balls $D' \subset D$ in $\mathbb{C}$ induces a homotopy equivalence between the respective Beilinson--Drinfeld Grassmannians $\mathrm{Gr}_{G, {D'}^n} \hookrightarrow \mathrm{Gr}_{G, D^n}$, for any positive integer $n$. We use a purely algebraic approach, showing that automorphisms of a complex smooth algebraic curve $X$ can be lifted to automorphisms of the associated Beilinson--Drinfeld Grassmannian. As a consequence, we obtain a stronger version of the usual homotopy invariance result: namely, the homotopies can be promoted to equivariant stratified isotopies, where "equivariant" refers to the action of the arc group $\mathrm{L}^+G$ and "stratified" refers to the stratification induced by the Schubert stratification of $\mathrm{Gr}_G$ and the incidence stratification of $\mathbb{C}^n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06222 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isotopy invariance and stratified $\mathbb{E}_2$-structure of the Ran Grassmannian Nocera, Guglielmo Porzio, Morena Algebraic Topology Algebraic Geometry 14D24, 14H60, 32S60, 55P10, 55P48 Let $G$ be a complex reductive group. A folklore result asserts the existence of an $\mathbb{E}_2$-algebra structure on the Ran Grassmannian of $G$ over $\mathbb{A}^1_{\mathbb{C}}$, seen as a topological space with the complex-analytic topology. The aim of this paper is to prove this theorem, by establishing a homotopy invariance result: namely, an inclusion of open balls $D' \subset D$ in $\mathbb{C}$ induces a homotopy equivalence between the respective Beilinson--Drinfeld Grassmannians $\mathrm{Gr}_{G, {D'}^n} \hookrightarrow \mathrm{Gr}_{G, D^n}$, for any positive integer $n$. We use a purely algebraic approach, showing that automorphisms of a complex smooth algebraic curve $X$ can be lifted to automorphisms of the associated Beilinson--Drinfeld Grassmannian. As a consequence, we obtain a stronger version of the usual homotopy invariance result: namely, the homotopies can be promoted to equivariant stratified isotopies, where "equivariant" refers to the action of the arc group $\mathrm{L}^+G$ and "stratified" refers to the stratification induced by the Schubert stratification of $\mathrm{Gr}_G$ and the incidence stratification of $\mathbb{C}^n$. |
| title | Isotopy invariance and stratified $\mathbb{E}_2$-structure of the Ran Grassmannian |
| topic | Algebraic Topology Algebraic Geometry 14D24, 14H60, 32S60, 55P10, 55P48 |
| url | https://arxiv.org/abs/2509.06222 |