nLab fixed point space

Idea

General

Generally, given some kind of space equipped with the action of a group, the locus of fixed points of the action may form a suitable sub-space: the fixed point space.

For topological $G$-spaces

Specifically, given a topological group $G$ and a topological G-space, its fixed point space is the set of the set-theoretic fixed points of the $G$-action, equipped with the subspace topology.

For more see at topological G-space the section Change of groups and fixed loci.

In equivariant homotopy theory

The statement of Elmendorf's theorem is essentially that the equivariant homotopy theory of topological $G$-spaces is equivalently encoded in their systems of $H$-fixed point spaces, as $H$ varies over closed subgroups of $G$.

In equivariant stable homotopy theory

In equivariant stable homotopy theory the concept of fixed point spaces branches into various closely related, but different concepts:

• fixed point spectra,

• geometric fixed point spectra

• categorical fixed point spectra?

In equivariant differential topology

In equivariant differential topology:

Proposition

(existence of $G$-invariant tubular neighbourhoods)

Let $X$ be a smooth manifold, $G$ a Lie group and $\rho \;\colon\; G \times X \to X$ a proper action by diffeomorphisms.

If $\Sigma \overset{\iota}{\hookrightarrow} X$ is a closed smooth submanifold inside the $G$-fixed locus

then $\Sigma$ admits a $G$-invariant tubular neighbourhood $\Sigma \subset U \subset X$.

Moreover, any two choices of such $G$-invariant tubular neighbourhoods are $G$-equivariantly isotopic.

(Kankaanrinta 07, theorem 4.4, theorem 4.6)

Proposition

(fixed loci of smooth proper actions are submanifolds)

Let $X$ be a smooth manifold, $G$ a Lie group and $\rho \;\colon\; G \times X \to X$ a proper action by diffeomorphisms.

Then the $G$-fixed locus $X^G \hookrightarrow X$ is a smooth submanifold.

(see also this MO discussion)

Proof

Let $x \in X^G \subset X$ be any fixed point. Since this is in particular a closed invariant submanifold, Prop. applies and shows that an open neighbourhood of $x$ in $X$ is $G$-equivariantly diffeomorphic to a linear representation $V \in RO(G)$. The fixed locus $V^G \subset V$ of that is hence diffeomorphic to an open neighbourhood of $x$ in $\Sigma$.

Remark

Without the assumption of proper action in Prop. the conclusion generally fails. See this MO comment for a counter-example.

Properties

Fixed point adjunction

For $G$ a topological group, consider the category of TopologicalGSpaces.

For $H \subset G$ any subgroup, consider

• the coset space $G/H \in Topological G Spaces$;

• the Weyl group $W_H(G) \coloneqq N_G(H)/H \in TopologicalGroups$.

Observing that for $X \in Topological G Spaces$ the $H$-fixed locus $X^H$ inherits a canonical action of $N(H)/H$, we have a functor

(1)$$Topological G Spaces \overset{ \;\;\; (-)^H \;\;\; }{\longrightarrow } Topological N(H)/H Spaces$$

Notice that $G$ acts canonically on the left of $G/H$, while $N(H)/H$ still acts from the right (both by group multiplication on representatives):

$$\array{ G/H \times N(H)/H &\overset{}{\longrightarrow}& G/H \\ \big( g H, n H \big) &\mapsto& g H n H \mathrlap{ \,=\, g n \underset{H}{\underbrace{n^{-1} H n}} H \,=\, g n H } }$$

Therefore there exists a functor in the other direction:

$$\array{ Topological N(H)/H Spaces & \overset{ \;\;\; G/H \times_{N(H)/H} (-) \;\;\; }{\longrightarrow} & Topological G Spaces } \,.$$

Proposition

(passage to fixed loci is a right adjoint)
These are adjoint functors, with the $H$-fixed locus functor (1) being the right adjoint:

(2)$$Topological G Spaces \underoverset { \underset{ (-)^H }{\longrightarrow} } { \overset{ G/H \times_{N(H)/H} (-) }{ \longleftarrow } } {\;\;\;\;\;\;\; \bot \;\;\;\;\;\;\;} Topological N(H)/H Spaces \,.$$

Proof

To see the hom-isomorphism characterizing this adjunction, consider for $X \in Topological N(H)/H Spaces$ and $Y \in Topological G Spaces$ a $G$-equivariant continuous function

$$G/H \times_{N(H)/H} Y \overset{ \;\;\; f \;\;\; }{\longrightarrow} X \,.$$

This restricts to an $N(H)$-equivariant function on the $N(H)$-topological subspace

$$\array{ Y &\overset{\;\;\;}{\hookrightarrow}& G/H \times_{N(H)/H} Y \\ y &\mapsto& \big[ e H , y \big] }$$

Since $Y$ is a fixed locus for $H \subset N(H)$, by equivariance this restriction has to factor through the $H$-fixed locus $X^H$ of $X$:

$$\array{ Y &\subset& G/H \times_{N(H)/H} Y \\ {}^{\mathllap{ \tilde f }} \big\downarrow && \big\downarrow {}^{\mathrlap{f}} \\ X^H &\subset& X }$$

But given that and since every other point of $G/H \times_{N(H)/H} Y$ is an image under the $G$-action of a point in $Y \subset G/H \times_{N(H)/H}$, this restriction $\tilde f$ already determines $f$ uniquely.

Since this construction is manifestly natural in $Y$ and $X$, we have a natural bijection $f \leftrightarrow \tilde f$, which establishes the hom-isomorphism for the pair of adjoint functors in (2).

Remark

The adjunction (2) factors as:

$$Topological G Spaces \underoverset { \underset{ }{\longrightarrow} } { \overset{ G \times_{N(H)} (-) }{ \longleftarrow } } {\;\;\;\;\;\;\; \bot \;\;\;\;\;\;\;} Topological N(H) Spaces \underoverset { \underset{ (-)^H }{ \longrightarrow } } { \overset{ N(H)/H \times_{N(H)/H} (-) }{ \longleftarrow } } {\;\;\;\;\;\;\; \bot \;\;\;\;\;\;\;} Topological N(H)/H Spaces \,.$$

Here the functor on the top right, $N(H)/H \times_{N(H)/H} (-)$, is the identity on the underlying topological spaces, but extends the action from $N(H)/H$ to $N(H)$, namely through the projection homomorphims $N(H) \to N(H)/H$.

For more on this see at Topological G-space – Fixed loci with residual Weyl gorup action:

Related concepts

• fixed point group

• homotopy fixed points

• Elmendorf's theorem

• equivariant homotopy theory

• equivariant differential topology