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edit approved Feb 21, 2018 at 9:37
M.G.
edit approved Feb 21, 2018 at 9:37
M.G.
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I'm trying to reconcile two results on the classification of principal bundles. First, we have Prin_G(X)$\mathrm{Prin}_G(X)$ (the equivalence classes of G$G$-bundles on X$X$) is isomorphic to H^1(X; G)$H^1(X;G)$ (the first Cech cohomology group of X -- I'm taking G$G$ to be abelian). Second, we have Prin_G(X)$\mathrm{Prin}_G(X)$ is isomorphic to [X,BG]$[X,BG]$, the set of homotopy equivalences of maps from X$X$ to BG$BG$, the classifying space of G$G$. If we now take G=R$G=\mathbb{R}$, 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$BG$ to be a point). How do I reconcile these? Does one result not apply in this case?

Thanks in advance!

I'm trying to reconcile two results on the classification of principal bundles. First, we have 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 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=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!

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!

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oggius
oggius
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Classification of principal bundles

I'm trying to reconcile two results on the classification of principal bundles. First, we have 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 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=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!