orbit projection; proper $G$-manifold; fibration; quasifibration
The orbit projection $\pi \: M \to M/G$ of a proper $G$-manifold $M$ is a fibration if and only if all points in $M$ are regular. Under additional assumptions 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.
Consequently, using the fact that G G/H, G G/K are fibrations, we obtain the commuting diagram n+1 (G) n+1 (G) // n+1 (G/H) // n+1 (G/K) // n (H) // n (K) // n (G) // n (G) : n (G/H) . n (G/K)