nLab slice theorem

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

($H$-slice)
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):

     (1)$$\array{ G \times_H S & \overset {\simeq} { \longrightarrow } & G \cdot S \\ [g, s] & \mapsto & g \cdot s }$$

  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 $x$.

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

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)

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

• $G$ be the topological group underlying a Lie group,

• $X$ be a locally compact Hausdorff space,

• $G \times X \overset{\rho}{\to} X$ be a proper action.

Then for every point $x \in X$ there exists a slice through $x$ (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 $X$ be locally compact, in which case the definition of “proper action” needs to be strengthened (“Palais proper action”, Palais 61, Def. 1.2.2). Under the assumption of local compactness, Palais’ more general statement reduces as above, see Karppinen 2016, Rem. 5.2.4.

When the group $G$ is compact then the condition on the $G$-space $X$ may be relaxed:

Proposition

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

Let

• $G$ be a compact topological group

• $X$ a completely regular topological space.

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

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

Examples and applications

General

Example

($G$-slice through $G$-fixed point)
If a point $x \in X$ in a topological G-space $X$ is fixed by all of $G$, so that $Stab_G(x) \,=\, G$, then $X$ itself is a $G$-slice through $x$ (Def. ), since we trivially have $G \times_G X \,\simeq\, X$ and $X \underset{open}{\subset} X$.

Example

(slices through points in orthogonal representation)
For $n \in \mathbb{N}$ consider the defining group action of the orthogonal group $G \coloneqq O(n+1)$ on the Cartesian space $\mathbb{R}^{n+1}$.

Then two cases of stabilizer groups appear:

1. the origin $0 \,\in\, \mathbb{R}^{n+1}$ is fixed by all of $O(n+1)$, and a $Stab_G(0) = O(n+1)$-slice through origin is given by all of $\mathbb{R}^{n+1}$ (by Ex. ) or by any open ball around it;

2. for every other point $x \in \mathbb{R}^{n+1} \setminus \{0\}$ the stabilizer subgroup is $G_x = O(n)$ and the coset space $G/G_x = S^n$ is the n-sphere (see there).

The ray $R_X \,\coloneqq\, \{c \cdot x \vert c \in \mathbb{R}_{\gt 0}\} \subset \mathbb{R}^{n+1} \setminus \{0\}$ has orbit the complement $\mathbb{R}^{n+1} \setminus \{0\}$ and is thus an $O(n)$-slice through $x$ (Def. ) as exhibited by the radial quotient map

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

$$R_x \times_{O(n)} O(n+1) \;\simeq\; \mathbb{R}^{n+1} \setminus \{0\} \,.$$

(also Karppinen 2016, Ex. 6.1.10)

Example

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.

Principal bundles

Proposition

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

1. the action by $G$ is free,

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

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

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

Proof

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

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

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

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

Setting

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

observe that this implies:

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

and this implies the claim:

1. $U_x \,\subset\, P/G$ is an open subset

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

2. $U_x$ satisfies the required condition (2)

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

(In fact, the slice $S_x$ we used in this argument gives the function $U_x \xrightarrow{\;\sim\;} S_x \xhookrightarrow{\;} P$ 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 $P$ such that

1. $G$ carries the structure of a Lie group,

2. $P$ is a locally compact Hausdorff space,

3. the action by $G$ is free and proper.

Then the quotient space coprojection $P \xrightarrow{q} P/G$ is a $G$-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 $G$-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).

Related concepts

• topological G-space, G-manifold

• compact Lie group

• equivariant bundle

• equivariant differential topology

• equivariant Tietze extension theorem

References

Original discussion:

• Andrew Gleason, Spaces With a Compact Lie Group of Transformations, Proceedings of the American Mathematical Society Proceeding, Vol. 1, No. 1 (Feb., 1950), pp. 35-43 (jstor:2032430, doi:10.2307/2032430)

• Deane Montgomery, Chung Tao Yang, The existence of a slice, Ann. of Math. 65 (1957), 108-116 (jstor:1969667, doi:10.2307/1969667)

• George Mostow, Equivariant embeddings in Euclidean space, Ann. of Math. 65 (1957), 432-446 (jhir:1774.2/46183, pdf scan)

• Richard Palais, Slices and equivariant embeddings, chapter VIII in: Armand Borel (ed.), Seminar on Transformation Groups, Annals of Mathematics Studies 46, Princeton University Press 1960 (jstor:j.ctt1bd6jxd)

• Richard Palais, Section 1.7 of: The classification of $G$-spaces, Memoirs of the AMS, 36, 1960 (ISBN:978-0-8218-9979-3 pdf, pdf)

• Richard Palais, On the Existence of Slices for Actions of Non-Compact Lie Groups, Annals of Mathematics Second Series, Vol. 73, No. 2 (Mar., 1961), pp. 295-323 (jstor:1970335, doi:10.2307/1970335, pdf)

• Glen Bredon, Section II.4 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN:9780080873596, pdf)

Review:

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

See also:

• Tammo tom Dieck, Section I.5 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

• Sergey Antonyan, Characterizing slices for proper actions of locally compact groups, Topology and its Applications Volume 239, 15 April 2018, Pages 152-159 (arXiv:1702.08093, doi:10.1016/j.topol.2018.02.026)

Discussion for G-CW complexes:

• Bernardo Uribe, Wolfgang Lück, Section 7 of: Equivariant principal bundles and their classifying spaces, Algebr. Geom. Topol. 14 (2014) 1925-1995 [arXiv:1304.4862, doi:10.2140/agt.2014.14.1925]