I’m on the lookout for an instance of a topological group $G$ performing by homeomorphisms on a metric house $X$ such that the orbit map $Xto X/G$ would not have the trail lifting property, that’s, there’s a path in $X/G$ that can not be lifted to $X$.
Non-example: If $G$ is a compact Lie group, then the orbit map has the trail lifting property (the truth is, there’s a a slice). Palais gave a generalization to correct Lie group actions.
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