Browsing by Subject "Group cohomology"

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    Bockstein closed 2-group extensions and cohomology of quadratic maps
    (Elsevier, 2012-05-01) Pakianathan, J.; Yalçın, E.
    A central extension of the form E: 0 → V→ G→ W→ 0, where V and W are elementary abelian 2-groups, is called Bockstein closed if the components qi∈H *(W,F 2) of the extension class of E generate an ideal which is closed under the Bockstein operator. In this paper, we study the cohomology ring of G when E is a Bockstein closed 2-power exact extension. The mod-2 cohomology ring of G has a simple form and it is easy to calculate. The main result of the paper is the calculation of the Bocksteins of the generators of the mod-2 cohomology ring using an Eilenberg-Moore spectral sequence. We also find an interpretation of the second page of the Bockstein spectral sequence in terms of a new cohomology theory that we define for Bockstein closed quadratic maps Q:. W→ V associated to the extensions E of the above form. © 2012 Elsevier Inc.
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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.