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Peer-reviewedCC BYJournal of Topology

Higher representation stability for ordered configuration spaces

  • Quoc P. Ho1
  1. 1Department of Mathematics The Hong Kong University of Science and Technology (HKUST) Hong Kong China
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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 . First, we provide an iterative procedure to study higher representation stability using actions coming from the cohomology of 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. Second, when certain ‐operations on the cohomology of 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 ‐modules arising from the cohomology of configuration spaces of open manifolds and, moreover, resolves the various conjectures of Miller–Wilson under these conditions.

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Cite

BibTeXRIS
BibTeX
@article{ho2026higher,
  title = {{Higher representation stability for ordered configuration spaces}},
  author = {Quoc P. Ho},
  journal = {Journal of Topology},
  year = {2026},
  month = jul,
  doi = {10.1112/topo.70092},
  url = {https://doi.org/10.1112/topo.70092},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Higher representation stability for ordered configuration spaces
AU  - Quoc P. Ho
PY  - 2026
DA  - 2026/07/29
JO  - Journal of Topology
DO  - 10.1112/topo.70092
UR  - https://doi.org/10.1112/topo.70092
AB  - 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 . First, we provide an iterative procedure to study higher representation stability using actions coming from the cohomology of 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. Second, when certain ‐operations on the cohomology of 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 ‐modules arising from the cohomology of configuration spaces of open manifolds and, moreover, resolves the various conjectures of Miller–Wilson under these conditions.
SN  - 1753-8416
SN  - 1753-8424
ER  -