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 H0 (G, A) and H^1 (G, A) and will find a six terms exact cohomology sequence involving H0 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 GnA 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.