When does $\pi:M\to M/G$ have homotopy lifting property?

Question

Let M be an n dimensional topological manifold, may be non-compact. Suppose there is an action of a group G on M, the orbits are closed, but may not be bounded. Consider the projection $\pi: M \to M/G$, do we have path lifting and homotopy lifting property? If not, what other conditions should we add?

I know when M is a smooth manifold, and G acts smoothly, freely, and properly, then $\pi$ is a smooth normal covering. But we consider manifold which may not be smooth and we only need lifting properties, the condition is too strict.

Consider the example $\pi: \mathbb{R}^3\to \mathbb{R}^3/Z_2$, $\mathbb{R}^3/Z_2$ is the cone over $\mathbb{R}P^2$, not a topological manifold. In this example, do we have path lifting and homotopy lifting property?

Answer 1

See

• Armin Rainer: Orbit projections as fibrations, Czechoslovak Math. J. 59 (2009), No. 2, 529--538, arXiv.

Abstract. The orbit projection $\pi : M → M/G$ of a proper $G$-manifold $M$ is a fibration if and only if all points in $M$ are regular. Under additional assump- tions we show that $\pi$ is a quasifibration if and only if all points are regular. We get a full answer in the equivariant category: $\pi$ is a $G$-quasifibration if and only if all points are regular.