Let $G$ be a Lie group and let $G^0$ be the identity component. It is known that $G^0$ is open and closed in $G$ and that $G/G^0$ is discrete. My question: Is it true that $G=G^0\times G/G^0$? Or equivalently, is the principal bundle $G\to G/G^0$ trivial?
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1See: https://mathoverflow.net/a/124622 – Gabe Conant Jun 26 '19 at 20:35
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@GabeConant What do you think about this answer https://math.stackexchange.com/q/316386 ? – Jun 26 '19 at 21:52