Framed configuration spaces and exotic spheres
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914578725601280 |
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| author | Krannich, Manuel Kupers, Alexander Mezher, Fadi |
| author_facet | Krannich, Manuel Kupers, Alexander Mezher, Fadi |
| contents | We determine when an exotic sphere $Σ$ of dimension $d\not \equiv 1 (4)$ can be detected through the homotopy type of its truncated Disc-presheaf. The latter records the diagram of framed configuration spaces of bounded cardinality in $Σ$ with natural point-forgetting and -splitting maps between them, and it gives rise to the finite stages in Goodwillie--Weiss' embedding calculus tower. Our proof involves three ingredients that could be of independent interest: a gluing result for Disc-presheaves of manifolds divided into two codimension zero submanifolds, a version of Atiyah duality in the context ofDisc-presheaves, and a computation of the finite residual of the mapping class group of the connected sums $\sharp^g(S^{2k+1}\times S^{2k+1})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Framed configuration spaces and exotic spheres Krannich, Manuel Kupers, Alexander Mezher, Fadi Algebraic Topology Geometric Topology 11F06, 18F50, 20E26, 55R80, 57R40, 57R60 We determine when an exotic sphere $Σ$ of dimension $d\not \equiv 1 (4)$ can be detected through the homotopy type of its truncated Disc-presheaf. The latter records the diagram of framed configuration spaces of bounded cardinality in $Σ$ with natural point-forgetting and -splitting maps between them, and it gives rise to the finite stages in Goodwillie--Weiss' embedding calculus tower. Our proof involves three ingredients that could be of independent interest: a gluing result for Disc-presheaves of manifolds divided into two codimension zero submanifolds, a version of Atiyah duality in the context ofDisc-presheaves, and a computation of the finite residual of the mapping class group of the connected sums $\sharp^g(S^{2k+1}\times S^{2k+1})$. |
| title | Framed configuration spaces and exotic spheres |
| topic | Algebraic Topology Geometric Topology 11F06, 18F50, 20E26, 55R80, 57R40, 57R60 |
| url | https://arxiv.org/abs/2509.19074 |