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 $G_{n}'s$?