Cohomology of infinite groups realizing fusion systems

Gündoğan, Muhammed Said; Yalcin, Ergun

doi:10.1007/s40062-019-00240-5

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Fusion Systems Realized By Finite Groups2

The Normalizer Decomposition For O C Pfq2

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2021

2024

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Cited by 3 publications

(4 citation statements)

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“…realizing F (see [16,Ex 4.3]). The subgroup a 2 ∼ = Z is central in G and G/ a 2 is isomorphic to the symmetric group S 3 .…”

Section: Fusion Systems Realized By Finite Groupsmentioning

confidence: 99%

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Higher limits over the fusion orbit category

Yalcin

2022

Advances in Mathematics

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“…realizing F (see [16,Ex 4.3]). The subgroup a 2 ∼ = Z is central in G and G/ a 2 is isomorphic to the symmetric group S 3 .…”

Section: Fusion Systems Realized By Finite Groupsmentioning

confidence: 99%

“…H * (P ; F 3 ) ∼ = H * (S 3 ; F 3 ) = H * (G; F 3 ), since H n−1 (S 3 ; F 3 ) = 0 for n = 0, 1 mod 4. Hence this is an example of the situation where the cohomology of the infinite group G realizing the fusion system F is not isomorphic to the cohomology of F (see [16]).…”

Section: Fusion Systems Realized By Finite Groupsmentioning

confidence: 99%

Higher limits over the fusion orbit category

Yalcin

2022

Advances in Mathematics

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“…Let F be the fusion system of the symmetric group S 3 at prime 3. If we apply the Leary-Stancu construction to F, we find the infinite group G " xb, a | b 3 " 1, aba ´1 " b 2 y -C 3 ¸Z realizing F (see [16,Ex 4.3]). The subgroup xa 2 y -Z is central in G and G{xa 2 y is isomorphic to the symmetric group S 3 .…”

Section: The Normalizer Decomposition For O C Pfqmentioning

confidence: 99%

“…We have H ˚pF ; F 3 q " lim P POpF q H ˚pP ; F 3 q -H ˚pS 3 ; F 3 q ‰ H ˚pG; F 3 q, since H n´1 pS 3 ; F 3 q ‰ 0 for n " 0, 1 mod 4. Hence this is an example of the situation where the cohomology of the infinite group G realizing the fusion system F is not isomorphic to the cohomology of F (see [16]).…”

Section: The Normalizer Decomposition For O C Pfqmentioning

confidence: 99%

Higher limits over the fusion orbit category

Yalcin¹

2021

Preprint

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The fusion orbit category FCpGq of a discrete group G over a collection C is the category whose objects are the subgroups H in C, and whose morphisms H Ñ K are given by the G-maps G{H Ñ G{K modulo the action of the centralizer group CGpHq. We show that the higher limits over FCpGq can be computed using the hypercohomology spectral sequences coming from the Dwyer G-spaces for centralizer and normalizer decompositions for G.If G is the discrete group realizing a saturated fusion system F, then these hypercohomology spectral sequences give two spectral sequences that converge to the cohomology of the centric orbit category O c pFq. This allows us to apply our results to the sharpness problem for the subgroup decomposition of a p-local finite group. We prove that the subgroup decomposition for every p-local finite group is sharp (over F-centric subgroups) if it is sharp for every p-local finite group with nontrivial center. We also show that for every p-local finite group pS, F, Lq, the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp.

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LHS-spectral sequences for regular extensions of categories

Yalçın

2024

J. Homotopy Relat. Struct.

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Cohomology of infinite groups realizing fusion systems

Cited by 3 publications

References 18 publications

Higher limits over the fusion orbit category

Higher limits over the fusion orbit category

Higher limits over the fusion orbit category

LHS-spectral sequences for regular extensions of categories

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