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

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 $(A,\mu)$, where $\mu:A\rightarrow R$ is a continuous homomorphism. Let $Der_{c}(G,(A,\mu))$ be the set of all $(\alpha,r)$ such that $\alpha:G\rightarrow A$ is a continuous crossed homomorphism and $\mu\alpha(g)=r^{g}r^{-1}$. 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.