Is there anything special about principal $G$-bundles, such that $G$- abelian group? Are there some interesting consequenes of this property, which do not take place in general?



Abelian principal bundles
In easy detail, the classifying space $BG$ for principal $G$-bundles ($G$ a topological abelian group) is itself a topological abelian group, just because the usual classifying space functor preserves products and the multiplication and inverse on $G$ are continuous homomorphisms. Since the set of principal $G$-bundles over any $X$ (assuming CW homotopy type) is in natural bijective correspondence with the set of homotopy classes of maps $X\to BG$, it naturally forms an abelian group. If $G$ is discrete, $BG$ is a $K(G,1)$ and this set is $H^1(X;G)$. In general, one can define continuous cohomology so that the conclusion remains true.


