On the motivic description of truncated fundamental group rings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916298500341760 |
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| author | Looijenga, Eduard |
| author_facet | Looijenga, Eduard |
| contents | A topological theorem that appears in a paper by Deligne-Goncharov (and which they attribute to Beilinson) states the following. Let $(X,*)$ be a path connected pointed space with a reasonable topology and denote by $I$ the augmentation ideal of its fundamental group ring. Then for every field F and positive integer n, the space of F-valued linear forms on $ I/I^{n+1}$ is naturally isomorphic to $H^n(X^n,X(n,*); F)$, where $X(n,*)$ is an explicitly defined subspace of $X^n$.
We here construct a simple isomorphism between $I/I^{n+1}$ and $H_n(X^n,X(n,*); \mathbf{Z})$ and express the maps that define the Hopf algebra structure on the $I$-adic completion of the fundamental group ring of $(X,*)$ in these terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the motivic description of truncated fundamental group rings Looijenga, Eduard Algebraic Geometry Geometric Topology A topological theorem that appears in a paper by Deligne-Goncharov (and which they attribute to Beilinson) states the following. Let $(X,*)$ be a path connected pointed space with a reasonable topology and denote by $I$ the augmentation ideal of its fundamental group ring. Then for every field F and positive integer n, the space of F-valued linear forms on $ I/I^{n+1}$ is naturally isomorphic to $H^n(X^n,X(n,*); F)$, where $X(n,*)$ is an explicitly defined subspace of $X^n$. We here construct a simple isomorphism between $I/I^{n+1}$ and $H_n(X^n,X(n,*); \mathbf{Z})$ and express the maps that define the Hopf algebra structure on the $I$-adic completion of the fundamental group ring of $(X,*)$ in these terms. |
| title | On the motivic description of truncated fundamental group rings |
| topic | Algebraic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2403.03748 |