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

(HSlice)
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):

      (1)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, recalled as Karppinen 2016, Thm. 6.2.7.

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). Under the assumption of local compactness, Palais’ more general statement reduces as above, see Karppinen 2016, Rem. 5.2.4.

When the group GG is compact then the condition on the GG-space XX may be relaxed:

Proposition

(existence of local slices for compact group actions on completely regular spaces)

Let

and G×XρXG \times X \overset{\rho}{\to} X any continuous action.

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

This is due to Mostow 1957, Thm. 2.1, reproved as Palais 60b, Cor. 1.7.19. It is also a special case of Palais 61, Prop. 2.3.1, using that compact group actions on completely regular spaces are “Cartan actions” in the sense of Palais (by Karppinen 2016, Prop. 5.1.7).

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

Examples and applications

General

Example

If time evolution on some Lorentzian manifold is given as an 1\mathbb{R}^1-action with timelike flow lines, then slices (“1-slices”) for this action are known as Cauchy surfaces.

Principal bundles

Proposition

(slice theorem implies that free quotient is principal bundle)
Consider a topological G-space PP such that

  1. the action by GG is free,

  2. there exist slices through each pPp \in P (Def. ).

Then the quotient space coprojection PqP/GP \xrightarrow{\;q\;} P/G is a locally trivial fiber bundle, in fact a GG-principal bundle.

(Palais 1961, p. 315, Sec. 4.1)
Proof

We need to show that for each point xP/Gx \in P/G there exists an open neighbourhood xU xP/Gx \,\in\, U_x \,\subset\, P/G such that its preimage under the coprojection is equivariantly homeomorphic to its product space with GG:

(2)G×U xq 1(U x)GAct(TopSp). G \times U _x \;\simeq\; q^{-1}(U_x) \;\;\;\;\;\; \in \; G Act(TopSp) \,.

Now, picking any preimage x^q 1({x})P\hat x \,\in\, q^{-1}\big(\{x\}\big) \,\subset\, P, there exists, by assumption, a G x^G_{\hat x}-slice S x^PS_{\hat x} \,\subset\, P through an open neighbourhood x^U^ x^P\hat x \in \widehat{U}_{\hat x} \,\subset\, P. But since the GG-action is assumed to be free, the stabilizer subgroup G x^1G_{\hat x} \,\simeq\, 1 is necessarily the trivial group, so that the defining property (1) of the slice is:

(3)G×S x^ U^ x^ openP (g,p) gp. \array{ G \times S_{\hat x} & \xrightarrow{ \;\; \sim \;\; } & \widehat{U}_{\hat x} & \underset{open}{\subset} \; P \\ (g, p) &\mapsto& g \cdot p \mathrlap{\,.} }

Setting

(4)U xU^ x^/GP/G, U_x \;\coloneqq\; \widehat{U}_{\hat x}/G \;\subset\; P/G \,,

observe that this implies:

q 1(U x)U^ x^. q^{-1}(U_x) \;\simeq\; \widehat{U}_{\hat x} \,.

and this implies the claim:

  1. U xP/GU_x \,\subset\, P/G is an open subset

    (since U^ x^P\widehat U_{\hat x} \,\subset\, P is open and using the definition of the quotient topology);

  2. U xU_x satisfies the required condition (2)

    (since this is now the slice condition (3)),

(In fact, the slice S xS_x we used in this argument gives the function U xS xP U_x \xrightarrow{\;\sim\;} S_x \xhookrightarrow{\;} P which is the local section that exhibits the local trivialization.)

As a corollary:

Proposition

(free and proper Lie group actions on locally compact Hausdorff spaces are locally trivial)
Consider a topological G-space PP such that

  1. GG carries the structure of a Lie group,

  2. PP is a locally compact Hausdorff space,

  3. the action by GG is free and proper.

Then the quotient space coprojection PqP/GP \xrightarrow{q} P/G is a GG-principal bundle.

Proof

By Prop. the assumptions imply that through each point there exists a slice, so that the claim follows by Prop. .

Remark

Due to Prop. , some authors define a GG-principal bundle to be a free and proper action on a locally compact Hausdorff space, without mentioning local trivializability (e.g. Raeburn & Williams 1991, Def. 2.1).

References

Original discussion:

Review:

See also:

Last revised on September 22, 2021 at 14:14:37. See the history of this page for a list of all contributions to it.