Let we have  a complex of abelian topological or  lie groups  $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$  such that the image of $G_{n}$ is a closed subgroup of $G_{n+1}$. Then  we have a complex of fundamental groups $$\ldots \to \pi_{1}(G_{n})\to \pi_{1}(G_{n+1})\to \ldots$$

>Are there some standard theorems or results about a comparison betwee the cohomology of $\pi_{1}(G_{n})$  and the $\pi_{1}$ of  cohomolgy of $G_{n}$?