Classification of principal bundles I'm trying to reconcile two results on the classification of principal bundles. First, we have $\mathrm{Prin}_G(X)$ (the equivalence classes of $G$-bundles on $X$) is isomorphic to $H^1(X;G)$ (the first Cech cohomology group of X -- I'm taking $G$ to be abelian). Second, we have $\mathrm{Prin}_G(X)$ is isomorphic to $[X,BG]$, the set of homotopy equivalences of maps from $X$ to $BG$, the classifying space of $G$. If we now take $G=\mathbb{R}$, the real line viewed as an additive group, the first result seems to say we can have non-trivial bundles, while the latter seems to contradict that (since we can take $BG$ to be a point). How do I reconcile these? Does one result not apply in this case?
Thanks in advance!
 A: The cohomology group $H^1(X,\mathbb{R})$ is the Cech cohomology group of the sheaf of differentiable functions over $X$, since there exist partition of unity, $H^1(X,\mathbb{R})=0.$ 
What is the right version of "partitions of unity implies vanishing sheaf cohomology"
A: Principal $(\mathbb R,+)$-bundles are indeed all trivial (Existence of a section suffices which exists for every fiber bundle with contractible fiber). You should consider principal $(\mathbb R\setminus\{0\},\cdot)$-bundles. 
A: The resolution, as others have said, is simply that $H^1_{cech}(X, \mathbb{R}) = 0$ if $\mathbb{R}$ is the additive group with the standard topology.
The reason you implicitly thought that this cohomology group is nonzero might have been due to the following "proof", which tripped me up in the past:

Suppose $H^1_{singular}(X, G)$ is nonzero.  This group classifies maps from $X$ to $K(G,1)$ up to homotopy, and $BG$ is a $K(G,1)$.  But $H^1_{cech}(X,G)$ classifies maps from $X$ to $BG$ up to homotopy, so the groups are the same and thus the latter is nonzero.

The flaw is in the step "$BG$ is a $K(G,1)$": this is only true if the topology on $G$ is discrete!
