nLab
slice theorem

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Representation theory

Contents

Idea

Given a group GG equipped with an action on some space XX, a slice through the GG-orbits in XX is a subspace SXS\hookrightarrow X such that XX is exactly exhausted by the GG-orbits of SS.

More generally, if HGH \subset G is a subgroup, then a slice through GG-orbits modulo HH, or simply an HH-slice, is a subspace SGS \subset G to which the HH-action on XX restricts, and such that the GG-orbits of SS modulo this HH-action on SS exactly exhaust XX.

In mathematics, specifically in equivariant differential topology, the terminology is traditionally used by default to refer to slices through open sub G-spaces UXU \subset X of a given ambient topological G-space XX.

Here by a slice theorem is traditionally meant a list of sufficient conditions which guarantee that suitable slices do exist in a given situation. This plays a central role for instance in the local triviality of equivariant bundles.

In physics one typically considers this for GG a gauge group, in which case one speaks of gauge slices and thinks of these as a choice of gauge fixing (see there for more).

Slices of group action

There is the following general abstract definition of slices via adjunctions (which however seems not to be made explicit in existing literature):

Definition

Let HGH \subset G be a subgroup inclusion such that restricting given actions (internal to a given ambient category) of GG to HH has a left adjoint G× H()G \times_{H} (-) (e.g. for topological G-spaces the topological induced action).

Then an HH-slice in a GG-action UU is an HH-subaction inclusion SiUS \overset{i}{\hookrightarrow} U whose induction/restriction-adjunct is an isomorphism:

G× HSi˜U. G \times_ H S \underoverset {\simeq} { \;\; \tilde i \;\; } {\longrightarrow} U \,.

Here are more traditional ways to say this:

Definition

Let GG be a topological group and XX a topological G-space.

For HGH \subset G a closed subgroup, a topological subspace SXS \subset X is called:

  • an HH-slice if

    1. SS is an HH-subspace;

    2. and HH is maximal with this property, in that the following map is an isomorphism (i.e. homeomorphism):

      G× HS GS [g,s] gs \array{ G \times_H S & \overset {\simeq} { \longrightarrow } & G \cdot S \\ [g, s] & \mapsto & g \cdot s }
    3. whose image GSXG\cdot S \subset X is open;

  • a slice through xXx \in X if

    1. xSXx \in S \subset X;

    2. SS is a G xG_x-slice in the above sense,

      for G xStab G(x)G_x \coloneqq Stab_G(x) the stabilizer group of xx.

(e.g. Bredon 72, Ch. II, Def. 4.1)

Alternatively but equivalently (as in Bredon 72, Ch. II, Thm. 4.4 (iv)):

Definition

Let GG be a topological group and XX a topological G-space.

For HGH \subset G a closed subgroup, a topological subspace SXS \subset X is called:

  • an HH-kernel if it is the preimage of the base point [H]G/H[H] \in G/H in the coset space under a GG-equivariant continuous function ff from the GG-orbit of SS:

    GSfG/HSf 1([H]) \underset{ G\cdot S \overset{f}{\to} G/H } { \exists } \; S \;\simeq\; f^{-1} \big( [H] \big)
  • an HH-slice if it is an HH-kernel and its orbit is an open subspace:

    GSopenX G \cdot S \underset{open} {\subset} X
  • a slice through xx if it is a G xG_x-slice for some xSXx \in S \subset X with stabilizer group G xStab G(x)HG_x \coloneqq Stab_G(x) \simeq H.

(Palais 61, Def. 2.1.1, recalled as Karppinen 16, Def. 6.1.1)

Example

For nn \in \mathbb{N} consider the defining action of the orthogonal group G=O(n+1)G = O(n+1) on the Cartesian space n+1\mathbb{R}^{n+1}. Then for every point 0x n+10 \neq x \in \mathbb{R}^{n+1} we have G x=O(n)G_x = O(n) and G/G x=S nG/G_x = S^n the n-sphere.

The ray {cx|c >0} \{c \cdot x \vert c \in \mathbb{R}_{\gt 0}\} \subset has orbit the complement of the origin and is thus a slice through xx as exhibited by the radial quotient map

n+1{0}S n=O(n+1)/O(n). \mathbb{R}^{n+1} \setminus \{0\} \longrightarrow S^n = O(n+1)/O(n) \,.

Properties

Existence of slices

A slice theorem is a statement of sufficient conditions such that there is a slice through each point of a given topological G-space.

Proposition

(existence of local slices for proper actions on locally compact spaces)
Let

Then for every point xXx \in X there exists a slice through xx (Def. ).

This is due to Palais 61, Prop. 2.3.1.

Remark

The thrust of Palais 61 is to state Prop. without the assumption that XX be locally compact, in which case the definition of “proper action” needs to be strengthened (“Palais proper action”, Palais 61, Def. 1.2.2).

For smooth G-manifolds the HH-space SS may be taken to be a linear representation (e.g. tomDieck 87, Thm. 5.6).

Examples

References

Original discussion:

Review:

See also:

Last revised on April 13, 2021 at 06:42:54. See the history of this page for a list of all contributions to it.