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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 $Gltimes A$ 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, $H^{1}(G,A)=1$.
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In this paper we introduce a new definition of the first non-abelian cohomology of topological groups. We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-abelian cohomology of a topological group by the notion of a principle homogeneous space.
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 $(A,mu)$, where $mu:Arightarrow R$ is a continuous homomorphism. Let $Der_{c}(G,(A,mu))$ be the set of all $(alpha,r)$ such that $alpha:Grightarrow A$ is a continuous crossed homomorphism and $mualpha(g)=r^{g}r^{-1}$. We introduce a topology on $Der_{c}(G,(A,mu))$. We show that $Der_{c}(G,(A,mu))$ is a topological group, wherever $G$ and $R$ are locally compact. We define the first cohomology, $H^{1}(G,(A,mu))$, of $G$ with coefficients in $(A,mu)$ as a quotient space of $Der_{c}(G,(A,mu))$. Also, we state conditions under which $H^{1}(G,(A,mu))$ is a topological group. Finally, we show that under what conditions $H^{1}(G,(A,mu))$ is one of the following: $k$-space, discrete, locally compact and compact.
Let $A$ be an abelian topological $G$-module. We give an interpretion for the second cohomology, $H^{2}(G,A)$, of $G$ with coefficients in $A$. As a result we show that if $P$ is a projective topological group, then $H^{2}(P,A)=0$ for every abelian topological $P$-module $A$.
For an odd prime p the cohomology ring of an elementary abelian p-group is polynomial tensor exterior. We show that the ideal of essential classes is the Steenrod closure of the class generating the top exterior power. As a module over the polynomial algebra, the essential ideal is free on the set of Mui invariants.
We prove continuity results for abstract epimorphisms of locally compact groups onto finitely generated groups.
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