Let be $G$ a compact Lie group and $\pi: EG \rightarrow BG$ the associated universal bundle. If $X$ is a compact Hausdorff space there is a one-to-one correspondence between the equivalence classes of principal $G-$bundles on $X$ and the homotopy classes of maps from $X$ to $BG$. We have that this correspondence is given by associating to each map $f:X \rightarrow BG$ the induced bundle $f^*(EG)$. But how can I build explicitly the isomorphism $$ Prin_{G}(X) \rightarrow [X,BG] $$ using the properties of induced bundle?
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I don't have time to check it in detail but I believe you can follow the method of proof given on pages 29 and 30 of Allen Hatchers "Vector Bundles and K-Theory" book available online. This treats only the case of Vector bundles, classified by homotopy classes in $[X, BO]$ where $O$ is the direct limit of the orthogonal groups (or $[X,BU]$) for complex bundles, but you can look and see if extends to general compact Lie groups. – tharris Jun 03 '13 at 15:37
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cont: Hatcher shows that the map $[X, BO] \rightarrow Vect(X)$ is surjective by constructing an $f$ on the left for each bundle on the right, so this will tell you how to define your map in the reverse direction. – tharris Jun 03 '13 at 15:37