Browsing by Subject "Orbit category"

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    Equivariant Moore spaces and the Dade group
    (Elsevier, 2017) Yalçın, E.
    Let G be a finite p-group and k be a field of characteristic p. A topological space X is called an n-Moore space if its reduced homology is nonzero only in dimension n. We call a G-CW-complex X an n_-Moore G-space over k if for every subgroup H of G, the fixed point set XH is an n_(H)-Moore space with coefficients in k, where n_(H) is a function of H. We show that if X is a finite n_-Moore G-space, then the reduced homology module of X is an endo-permutation kG-module generated by relative syzygies. A kG-module M is an endo-permutation module if Endk(M)=M⊗kM⁎ is a permutation kG-module. We consider the Grothendieck group of finite Moore G-spaces M(G), with addition given by the join operation, and relate this group to the Dade group generated by relative syzygies. © 2017 Elsevier Inc.
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    Higher limits over the fusion orbit category
    (Elsevier, 2022-06-09) Yalçın, Ergün
    The fusion orbit category F‾C(G) of a discrete group G over a collection C is the category whose objects are the subgroups H in C, and whose morphisms H→K are given by the G-maps G/H→G/K modulo the action of the centralizer group CG(H). We show that the higher limits over F‾C(G) 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 Oc(F). 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 (S,F,L), the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp.
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    Relative group cohomology and the Orbit category
    (Taylor & Francis, 2014) Pamuk, S.; Yalçın, E.
    Let G be a finite group and ℱ be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative ℱ-projective resolution for ℤ when ℱ is the family of all subgroups H ≤ G with rk H ≤ rkG - 1. We answer this question negatively by calculating the relative group cohomology ℱH*(G, F{double-struck}2) where G = ℤ/2 × ℤ/2 and ℱ is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology ℱH*(G, M) can be calculated using the ext-groups over the orbit category of G restricted to the family ℱ. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group G and whose horizontal line at E 2 page is isomorphic to the relative group cohomology of G. © 2014 Copyright Taylor & Francis Group, LLC.