On Topological Structure of the First Non-abelian Cohomology of Topological Groups

2014/02/17 by H. Sahleh, Hossein Sahleh, Sahleh, Hossein +2
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Advanced Topology and Set Theory #math.GN #math.GR #msc:18G50 #msc:20J06 #msc:22A05

paper · pdf · doi:10.48550/arxiv.1402.4168

15 pages

arxiv created 2014/08/19 · arxiv updated 2014/08/20

Abstract

Let G, R and A be topological groups. Suppose that G and R act continuously on A, and G acts continuously on R. In this paper, we define a partially crossed topological G-R-bimodule (A,μ), where μ:A→ R is a continuous homomorphism. Let Der_c(G,(A,μ)) be the set of all (α,r) such that α:G→ A is a continuous crossed homomorphism and μα(g)=r^gr^-1. We introduce a topology on Der_c(G,(A,μ)). We show that Der_c(G,(A,μ)) is a topological group, wherever G and R are locally compact. We define the first cohomology, H¹(G,(A,μ)), of G with coefficients in (A,μ) as a quotient space of Der_c(G,(A,μ)). Also, we state conditions under which H¹(G,(A,μ)) is a topological group. Finally, we show that under what conditions H¹(G,(A,μ)) is one of the following: k-space, discrete, locally compact and compact.

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