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First Non-Abelian Cohomology of Topological Groups

Let $G$ be a topological group and $A$ a topological $G$-module (not necessarily abelian). In this paper, we define $H^{0}(G,A)$ and $H^{1}(G,A)$ and will find a six terms exact cohomology sequence involving $H^{0}$ and $H^{1}$. We will extend it to a seven terms exact sequence of cohomology up to dimension two. We find a criterion such that vanishing of $H^{1}(G,A)$ implies the connectivity of $G$. We show that if $H^{1}(G,A)=1$, 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, $H^{1}(G,A)=1$.