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Let $G$ be a Lie group acting smoothly and freely on a smooth manifold $M$. Suppose that the quotient space $M/G$ is a topological manifold. Do we have $$\dim(M/G)=\dim M-\dim G?$$

Notes: This question was posted here on MSE. If $G$ acts properly, then of course the answer is yes. More generally, Daniel Robert-Nicoud observed here that the answer is also yes if there is a smooth structure on $M/G$ for which the projection $M\to M/G$ is a smooth submersion.

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  • $\begingroup$ It should be true if $M$ is locally trivial over $M/G$ with the fibre $G$. Some sufficient conditions for this are contained in the book Homotopical Topology by Fomenko and Fuchs, page 107. $\endgroup$
    – William of Baskerville
    Commented Aug 31, 2016 at 17:05
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    $\begingroup$ One can relax properness to the notion used by Palais in his paper on slices of Lie group actions. One should then check if for free group actions Palais condition is equivalent to Hausdorfness of $M/G$ (there is a good chance that this is true since this holds for actions of discrete groups). R. Palais, On the existence of slices for actions of non-compact Lie groups. Ann. of Math. (2) 73 (1961) 295-323. $\endgroup$
    – Misha
    Commented Sep 1, 2016 at 3:58

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