First Non-Abelian Cohomology of Topological Groups

2014/12/22 by Hossein Sahleh, Sahleh, Hossein, Hossein Esmaili Koshkoshi +1
Mathematics · #20J06 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Primary 22A05 #Secondary 18G50 #math.GN #math.GR #msc:18G50 #msc:20J06 #msc:22A05

paper · pdf · doi:10.48550/arxiv.1412.7066

15 pages

arxiv created 2014/12/22 · arxiv updated 2014/12/23

Abstract

Let G be a topological group and A a topological G-module (not necessarily abelian). In this paper, we define H⁰(G,A) and H¹(G,A) and will find a six terms exact cohomology sequence involving H⁰ and H¹. We will extend it to a seven terms exact sequence of cohomology up to dimension two. We find a criterion such that vanishing of H¹(G,A) implies the connectivity of G. We show that if H¹(G,A)=1, then all complements of A in the semidirect product G\ltimes A are conjugate. Also as a result, we prove that if G is a compact Hausdorff group and A is a locally compact almost connected Hausdorff group with the trivial maximal compact subgroup then, H¹(G,A)=1.

Related

• First Non-abelian Cohomology of Topological Groups II
• On Topological Structure of the First Non-abelian Cohomology of Topological Groups
• Nonabelian cohomology with coefficients in Lie groups
• Partial cohomology of groups and extensions of semilattices of abelian groups
• The First Cohomology Group H¹(G,M)