Contents

# Contents

## Idea

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

More generally, if $H \subset G$ is a subgroup, then a slice through $G$-orbits modulo $H$, or simply an $H$-slice, is a subspace $S \subset G$ to which the $H$-action on $X$ restricts, and such that the $G$-orbits of $S$ modulo this $H$-action on $S$ exactly exhaust $X$.

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

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

Then an $H$-slice in a $G$-action $U$ is an $H$-subaction inclusion $S \overset{i}{\hookrightarrow} U$ whose induction/restriction-adjunct is an isomorphism:

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

Here are more traditional ways to say this:

###### Definition

Let $G$ be a topological group and $X$ a topological G-space.

For $H \subset G$ a closed subgroup, a topological subspace $S \subset X$ is called:

• an $H$-slice if

1. $S$ is an $H$-subspace;

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

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

• a slice through $x \in X$ if

1. $x \in S \subset X$;

2. $S$ is a $G_x$-slice in the above sense,

for $G_x \coloneqq Stab_G(x)$ the stabilizer group of $x$.

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

###### Definition

Let $G$ be a topological group and $X$ a topological G-space.

For $H \subset G$ a closed subgroup, a topological subspace $S \subset X$ is called:

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

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

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

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

###### Example

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

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

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

## 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 $x \in X$ there exists a slice through $x$ (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 $X$ 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 $H$-space $S$ may be taken to be a linear representation (e.g. tomDieck 87, Thm. 5.6).

## Examples

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

Original discussion:

Review:

• Sini Karppinen, The existence of slices in $G$-spaces, when $G$ is a Lie group, Helsinki 2016 (hdl:10138/190707)