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][1] on MSE. If $G$ acts properly, then of course the answer is yes. More generally, Daniel Robert-Nicoud observed [here][2] 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.


  [1]: http://math.stackexchange.com/q/1900098/39285
  [2]: http://math.stackexchange.com/a/1900227/39285