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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).
There is the following general abstract definition of slices via adjunctions (which however seems not to be made explicit in existing literature):
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:
Here are more traditional ways to say this:
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
$S$ is an $H$-subspace;
and $H$ is maximal with this property, in that the following map is an isomorphism (i.e. homeomorphism):
whose image $G\cdot S \subset X$ is open;
a slice through $x \in X$ if
$x \in S \subset X$;
$S$ 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)):
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 $S$:
an $H$-slice if it is an $H$-kernel and its orbit is an open subspace:
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)
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
A slice theorem is a statement of sufficient conditions such that there is a slice through each point of a given topological G-space.
(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.
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).
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, 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:
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)
Last revised on April 13, 2021 at 06:42:54. See the history of this page for a list of all contributions to it.