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Comparison of the Discrete and Continuous Cohomology Groups of a Pro-$p$ Group
Gustavo A. Fernandez-Alcober ; Ilya V. Kazachkov ; Vladimir N. Remeslennikov ; Peter Symonds ;
Date:	25 Jan 2007
Subject:	Group Theory
Abstract:	We address the following question. For which finitely generated pro-$p$ groups the comparison map $phi^2:H_{cont}^{2}(P,F_p) o H_{disc}{2}(P,F_p)$ is an isomorphism? We prove that if $P$ is not finitely presented then $phi^2$ is not surjective. Furthermore, if $P$ is finitely presented $phi^2$ is an isomorphism if and only if the comparison map $phi_2:H^{disc}_{2}(P, F_p) o H^{cont}_{2}(P, F_p)$ of second homology groups is an isomorphism. This is the content of Theorem A. The second main result of the paper is Theorem B, which gives an explicit construction of a cochain from the kernel of $phi^2$.
Source:	arXiv, math/0701737
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