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Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras
[Submitted on 1 Apr 2020 (v1), last revised 23 Jun 2026 (this ve · 2026-06-24 · via math updates on arXiv.org

Mathematics > Algebraic Topology

arXiv:2004.00252 (math)

[Submitted on 1 Apr 2020 (v1), last revised 23 Jun 2026 (this version, v5)]

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Abstract:Using factorization homology with coefficients in twisted commutative algebras (TCAs), we prove two flavors of higher representation stability for the cohomology of (generalized) configuration spaces of a scheme/topological space $X$. First, we provide an iterative procedure to study higher representation stability using actions coming from the cohomology of $X$ and prove that all the modules involved are finitely generated over the corresponding TCAs. More quantitatively, we compute explicit bounds for the derived indecomposables in the sense of Galatius-Kupers-Randal-Williams. Secondly, when certain $C_\infty$-operations on the cohomology of $X$ vanish, we prove that the cohomology of its configuration spaces forms a free module over a TCA built out of the configuration spaces of the affine space. This generalizes a result of Church-Ellenberg-Farb on the freeness of $\mathrm{FI}$-modules arising from the cohomology of configuration spaces of open manifolds and, moreover, resolves the various conjectures of Miller-Wilson under these conditions.

Submission history

From: Quoc Ho [view email]
[v1] Wed, 1 Apr 2020 06:41:14 UTC (54 KB)
[v2] Fri, 3 Apr 2020 13:47:44 UTC (56 KB)
[v3] Sun, 10 May 2020 11:51:41 UTC (57 KB)
[v4] Wed, 28 Oct 2020 15:11:12 UTC (58 KB)
[v5] Tue, 23 Jun 2026 01:40:17 UTC (67 KB)

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