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coudy
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Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundlebundles on $M$ with structural group $G$ in smooth category. And denote by $Bun(M,G)_{top}$ the set of all equivalent Principal bundlebundles on $M$ with structural group $G$ in the topological category when $G$ is regardseen as a topological group. Then we have a natural map $\varphi:Bun(M,G) \rightarrow Bun(M,G)_{top}$. IfIs $\varphi$ is a bijection and how to prove it?

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundle on $M$ with structural group $G$ in smooth category. And denote by $Bun(M,G)_{top}$ the set of all equivalent Principal bundle on $M$ with structural group $G$ in the topological category when $G$ is regard as a topological group. Then we have a natural map $\varphi:Bun(M,G) \rightarrow Bun(M,G)_{top}$. If $\varphi$ is a bijection and how to prove it?

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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edit approved Apr 17, 2015 at 18:17
user70655
edit approved Apr 17, 2015 at 18:17
user70655

fiber Fiber bundle in smooth category and topological category

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundle on $M$ with structural group $G$ in smooth category. And denote by $Bun(M,G)_{top}$ the set of all equivalent Principal bundle on $M$ with structural group $G$ in the topological category when $G$ is regard as a topological group. Then we have a natural map $\varphi:Bun(M,G) \rightarrow Bun(M,G)_{top}$. If $\varphi$ is a bijection and how to prove it  ?

fiber bundle in smooth category and topological category

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundle on $M$ with structural group $G$ in smooth category. And denote by $Bun(M,G)_{top}$ the set of all equivalent Principal bundle on $M$ with structural group $G$ in the topological category when $G$ is regard as a topological group. Then we have a natural map $\varphi:Bun(M,G) \rightarrow Bun(M,G)_{top}$. If $\varphi$ is a bijection and how to prove it  ?

Fiber bundle in smooth category and topological category

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundle on $M$ with structural group $G$ in smooth category. And denote by $Bun(M,G)_{top}$ the set of all equivalent Principal bundle on $M$ with structural group $G$ in the topological category when $G$ is regard as a topological group. Then we have a natural map $\varphi:Bun(M,G) \rightarrow Bun(M,G)_{top}$. If $\varphi$ is a bijection and how to prove it?

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syms
syms
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fiber bundle in smooth category and topological category

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundle on $M$ with structural group $G$ in smooth category. And denote by $Bun(M,G)_{top}$ the set of all equivalent Principal bundle on $M$ with structural group $G$ in the topological category when $G$ is regard as a topological group. Then we have a natural map $\varphi:Bun(M,G) \rightarrow Bun(M,G)_{top}$. If $\varphi$ is a bijection and how to prove it ?