slice theorem

Motivation

Statement

Let be a manifold with a proper group action by a topological group or Lie group for a general manifold or smooth manifold respectively. Consider , with stabilizer subgroup , and normal space . There exists a -equivariant diffeomorphism from the associated bundle to the manifold

that takes to .

Slice definition

Why is the above about slices?

Let act on a topological space . For a closed subgroup a subspace is a called an -slice if

1. is -invariant;

is an isomorphism (so a homeomorphism or diffeomorphism depending on the category); 3. is open in .

A subspace is called a slice through if

1. ;
2. is a -slice.

Note: This is similar, but not the same as a fundamental domain. Slices are nice ways to characterize/ index orbits up to stabilizer subgroups, while maintaining open subsets.

Tracking across trivializations

Consider the principal bundle with local trivialization

Then using the trivializations of the associated bundle, we have

Note that . By the orbit stabilizer theorem, . Therefore, we have that neighborhoods around the orbit look like the orbit times the normal space, which can be studied by looking at the action on the normal space.

Relation to tubular neighborhood

Consider the orbit submanifold (since the action is proper, the orbit is an embedded submanifold). Then, we can look at the normal bundle

Note that since acts on , we can take the lifted action on .

is the tangent space to the orbit which is -invariant. We can see this since the tangent space has the following identification:

using the fundamental vector fields.

Therefore, for an element in , we have

So the normal bundle then has an induced action.

The main idea of the slice theorem is the map

(where the orbit seen as the 0 section is sent to itself in ) is -equivariant.