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Setting: Let $M$ be a smooth manifold and $\{U_\alpha\}_{\alpha \in \mathcal{I}}$ a locally finite covering of open subsets. Furthermore let $G$ be a smooth Lie group. Now assume we are given a family of transition functions $$\phi_{\alpha\beta}\in \mathcal{C}^0(U_\alpha \cap U_\beta; G) \quad\text{for any}\quad \alpha,\beta \in \mathcal{I}$$ satisfying the cocyclicity conditions: $$ \phi_{\alpha\alpha} = \mathbb{1}_{U_\alpha} \\ \phi_{\alpha\beta} \phi_{\beta\gamma} = \phi_{\alpha\gamma}. $$ on the obvious domains. Now $P := \left( \coprod_{\alpha \in \mathcal{I}} U_\alpha \times G \right)\big/{\sim}$ with $$(x,g)_\alpha \sim (x, \phi_{\beta\alpha}(x)\cdot g)_\beta \quad\text{for any}\quad x \in U_\alpha \cap U_\beta$$ defines a $G$-principal fibre bundle over $M$ (in the category of topological spaces).

Question: Can we conclude that $P \rightarrow M$ is a smooth $G$-principal bundle over $M$? That is, a principal bundle in the category of smooth manifolds?

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    $\begingroup$ No, we can't. In general, there is no reason to expect continuous transition maps to yield a smooth bundle. $\endgroup$ Jan 13 '18 at 11:35
  • $\begingroup$ @AmitaiYuval I find it hard to come up with a counterexample. Do you have any in mind or do you just suspect this to hold true? When I posed the question, I was thinking about Chern-classes, which classify certain principal bundles over a manifold. They are topological invariant but over smooth base spaces we can (I think) realise every topological classe of a bundle by a smooth principal bundle. $\endgroup$ Jan 14 '18 at 21:53
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Proof of continuous principal bundle is equivalent to smooth principal bundle
So the only thing left to do is to prove that your construction gives out a continuous principal bundle.
The first condition of been a continuous principal bundle is: The right $G$ action is continuous. Which in this construction, $(U_\alpha\times G)\times G\to(U_\alpha\times G)$, $(x,g)h\mapsto (x,gh)$, Because of the definition of Lie groups, it must be smooth, thus continuous.
And the second condition says that base manifold $M$ must exist local trivial cover $U_\alpha$, and that is already given in your construction.
So we conclude the $G$-principle bundle given is a continuous $G$-principal bundle. $\square$

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