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 $\mu :A\to R$ is a continuous homomorphism. Let $De{r}_c(G,(A,\mu ))$ be the set of all $(\alpha ,r)$ such that $\alpha :G\to A$ is a continuous crossed homomorphism and $\mu \alpha (g)=r^g{r}^{-1}$. We introduce a topology on $De{r}_c(G,(A,\mu ))$. We show that $De{r}_c(G,(A,\mu ))$ 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 $De{r}_c(G,(A,\mu ))$. 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.

arXiv comment15 pages

References in corpus (1)

• First Non-abelian Cohomology of Topological Groups II