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I want to know the existence of global section of $\pi : M\rightarrow M/G$, where

$M$ is a Riemannian manifold with $G$-action.

For instance in case of $M=S^2$ and $G={\bf Z}_2$ there exists no global section.

This case is easy by considering continuity.

(1) But I cannot show the noexistence of global section of $\pi : S^3 \rightarrow S^3/S^1=S^2$.

(2) And if $G$-action on $M$ has a fixed point, then there exists a global section.

How can we show ?

Thank you in advance.

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    $\begingroup$ I have an answer for (1) : if there eixsts such global section, $S^2\rightarrow S^3 \rightarrow S^2$ is identity so that we have identity on $H_2(S^2)$ This is a contradiction since $H_2(S^3)=0$ $\endgroup$ – HK Lee Aug 4 '13 at 5:18
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Hint for #1: A principal bundle has a global section if and only if it is a trivial bundle.

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  • $\begingroup$ If $ \pi : P \rightarrow B$ has a section $s$ and if $f_U : \pi^{-1}(U)\rightarrow U\times G$ is given by $gs(b)\mapsto (b,g)$ so $f_V \circ (f_U)^{-1} =id$. $\endgroup$ – HK Lee Aug 4 '13 at 7:49

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