Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundles on $M$ with structural group $G$ in smooth category. And denote by $Bun(M,G)_{top}$ the set of all equivalent Principal bundles on $M$ with structural group $G$ in the topological category when $G$ is seen as a topological group. Then we have a natural map $\varphi:Bun(M,G) \rightarrow Bun(M,G)_{top}$. Is $\varphi$ a bijection and how to prove it?
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$\begingroup$ Crossposted on MSE. $\endgroup$– Michael AlbaneseCommented Apr 17, 2015 at 18:04
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$\begingroup$ @IgorBelegradek the space $BG$ is not a manifold, right? What do you mean by a smooth map into it? $\endgroup$– Fernando MuroCommented Apr 17, 2015 at 21:13
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1$\begingroup$ This should be true, at least if $G$ is compact, by the existence of approximations of $BG$ by finite dimensional smooth manifolds and smooth approximation of continuous functions. $\endgroup$– skupersCommented Apr 17, 2015 at 23:37
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$\begingroup$ By math.stackexchange.com/questions/298873 $\endgroup$– David Roberts ♦Commented Apr 18, 2015 at 0:54
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$\begingroup$ @skupers: your remark doesn't have any meaning unless you say which version of $BG$ you are using. $BG$ has many different (albeit homotopy equivalent) versions. $\endgroup$– John KleinCommented Apr 20, 2015 at 21:31
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