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Higher limits over the fusion orbit category via subgroup complexes

  • Ergun Yalcin
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Abstract

Let β„± be a saturated fusion system over a finite p-group S, and π’ͺc (β„±) denote the centric orbit category of β„±. Suppose that G is a discrete group realizing β„±, and π’ž is the collection of all p-centric subgroups in G. We prove that for every β„€(p) π’ͺ c (β„±)-module M, the higher limits lim * π’ͺc (β„±) M are isomorphic to the equivariant cohomology HG * (|π’ž| ; β„³ ), where β„³ is the G-equivariant coefficient system on |π’ž| induced by M. This is the fusion system version of a theorem by Grodal for higher limits over the p-orbit category π’ͺp(G) of a finite group G. As a consequence, for every fusion system β„±, we obtain a finite sequence of cohomology groups, involving the cohomology of β„±, which is exact if and only if the mod-p subgroup decomposition for β„± over β„±-centric subgroups is sharp.

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Cite

BibTeXRIS
BibTeX
@misc{yalcin2026higher,
  title = {{Higher limits over the fusion orbit category via subgroup complexes}},
  author = {Ergun Yalcin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40915},
  eprint = {2610.40915},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40915},
}
RIS
TY  - UNPB
TI  - Higher limits over the fusion orbit category via subgroup complexes
AU  - Ergun Yalcin
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40915
UR  - https://arxiv.org/abs/2610.40915
AB  - Let β„± be a saturated fusion system over a finite p-group S, and π’ͺ^c (β„±) denote the centric orbit category of β„±. Suppose that G is a discrete group realizing β„±, and π’ž is the collection of all p-centric subgroups in G. We prove that for every β„€_(p) π’ͺ ^c (β„±)-module M, the higher limits lim ^* _π’ͺ^c (β„±) M are isomorphic to the equivariant cohomology H_G ^* (|π’ž| ; β„³ ), where β„³ is the G-equivariant coefficient system on |π’ž| induced by M. This is the fusion system version of a theorem by Grodal for higher limits over the p-orbit category π’ͺ_p(G) of a finite group G. As a consequence, for every fusion system β„±, we obtain a finite sequence of cohomology groups, involving the cohomology of β„±, which is exact if and only if the mod-p subgroup decomposition for β„± over β„±-centric subgroups is sharp.
ER  -