Linear stable ranges for integral homotopy groups of configuration spaces
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910061551419392 |
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| author | Guès, Nicolas |
| author_facet | Guès, Nicolas |
| contents | We prove explicit linear stable ranges for the $\mathsf{FI}$-modules $\mathrm{Hom}(π_p \mathrm{Conf} M, \mathbb Z)$ and $\mathrm{Ext}(π_p \mathrm{Conf} M, \mathbb Z)$ with $\mathrm{Conf} M$ being the configuration co$\mathsf{FI}$-space of a $d$-dimensional manifold with $d \geq 3$. The proof of this result uses a homotopy-theoretic approach to representation stability for $\mathsf{FI}$-modules. This allows us to derive representation stability results from homotopy-theoretical statements, in particular the generalized Blakers-Massey theorem. We also generalize to $\mathsf{FI}_G$-modules and orbit configuration spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear stable ranges for integral homotopy groups of configuration spaces Guès, Nicolas Algebraic Topology We prove explicit linear stable ranges for the $\mathsf{FI}$-modules $\mathrm{Hom}(π_p \mathrm{Conf} M, \mathbb Z)$ and $\mathrm{Ext}(π_p \mathrm{Conf} M, \mathbb Z)$ with $\mathrm{Conf} M$ being the configuration co$\mathsf{FI}$-space of a $d$-dimensional manifold with $d \geq 3$. The proof of this result uses a homotopy-theoretic approach to representation stability for $\mathsf{FI}$-modules. This allows us to derive representation stability results from homotopy-theoretical statements, in particular the generalized Blakers-Massey theorem. We also generalize to $\mathsf{FI}_G$-modules and orbit configuration spaces. |
| title | Linear stable ranges for integral homotopy groups of configuration spaces |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2503.21556 |