Knowlpedia

Orbit space

The quotient of a G-manifold by the equivalence relation of lying in the same orbit.

Let GG be a acting smoothly on a manifold MM (see ).

The orbit space (or quotient space) is the set

M/G={G⋅x∣x∈M}M/G=\{G\cdot x \mid x\in M\}

of all , equipped with the quotient topology for the canonical projection

π:M→M/G,π(x)=G⋅x.\pi:M\to M/G,\quad \pi(x)=G\cdot x.
Smooth structure in the free and proper case

In general, M/GM/G need not be a manifold. A standard sufficient condition is:

  • If the action is and , then M/GM/G carries a unique smooth manifold structure such that π\pi is a smooth submersion.

In this situation, each orbit is embedded and diffeomorphic to GG, and π\pi exhibits MM as a principal homogeneous object for GG along the fibers (compare ).

Basic examples
  • If the action is , then M/GM/G is a single point.
  • If M=GM=G acts on itself by left translation, then all orbits are all of GG and again M/GM/G is a point; if GG acts by conjugation, orbit spaces encode conjugacy classes and are typically singular.