Let we have  a complex of abelian topological or  lie groups  $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$ 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 $\pi_{1}(G_{n})'s$  and the $\pi_{1}$ of  cohomolgy of $G_{n}'s$?