The cohomology of pro-p groups with a powerfully embedded subgroup

Minh, Pham Anh; Symonds, Peter

doi:10.1016/j.jpaa.2003.10.034

Cited by 4 publications

(3 citation statements)

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“…The mod-2 cohomology ring structure of 2-power exact groups has a simple form and it is not very difficult to obtain once certain algebraic lemmas are established. Similar calculations were given by Rusin [9, Lemma 8] and Minh-Symonds [4].…”

Section: Definition 51 a Central Extension Of The Formsupporting

confidence: 61%

“…In Section 5, we calculate the mod-2 cohomology ring of a Bockstein closed 2-power exact group G using the Lyndon-Hochschild-Serre spectral sequence associated to the extension E. This calculation is relatively easy and it is probably known to experts in the field (see, for example, [4] or [9]). The mod-2 cohomology ring of G has a very nice expression given by…”

Section: Introductionmentioning

confidence: 99%

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Bockstein closed 2-group extensions and cohomology of quadratic maps

Pakianathan

Yalcin

2012

Journal of Algebra

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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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Section: Definition 51 a Central Extension Of The Formsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Bockstein closed 2-group extensions and cohomology of quadratic maps

Pakianathan

Yalcin

2012

Journal of Algebra

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“…(This is an important ingredient in calculating the cohomology of these groups as rings and as modules over the Steenrod algebra. See for example [1] and [4]. )…”

Section: Introductionmentioning

confidence: 99%

Quadratic maps and Bockstein closed group extensions

Pakianathan

Yalcin²

2007

Trans. Amer. Math. Soc.

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Abstract. Let E be a central extension of the form 0 → V → G → W → 0 where V and W are elementary abelian 2-groups. Associated to E there is a quadratic map Q : W → V , given by the 2-power map, which uniquely determines the extension. This quadratic map also determines the extension class q of the extension in H 2 (W, V ) and an ideal I(q) in H 2 (G, Z/2) which is generated by the components of q. We say that E is Bockstein closed if I(q) is an ideal closed under the Bockstein operator.We find a direct condition on the quadratic map Q that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic mapOn the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension 0In this situation, one may write β(q) = Lq for a "binding matrix" L with entries in H 1 (W, Z/2). We find a direct way to calculate the module structure of M in terms of L. Using this, we study extensions where the lattice M is diagonalizable/triangulable and find interesting equivalent conditions to these properties.

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