nLab
fixed point space

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

Contents

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 GG-spaces

Specifically, given a topological group GG and a topological G-space, its fixed point space is the set of the set-theoretic fixed points of the GG-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 G -spaces is equivalently encoded in their systems of HH-fixed point spaces, as HH varies over closed subgroups of GG.

In equivariant stable homotopy theory

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

In equivariant differential topology

In equivariant differential topology:

Proposition

(existence of GG-invariant tubular neighbourhoods)

Let XX be a smooth manifold, GG a Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

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

then Σ\Sigma admits a GG-invariant tubular neighbourhood ΣUX\Sigma \subset U \subset X.

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

(Kankaanrinta 07, theorem 4.4, theorem 4.6)

Proposition

(fixed loci of smooth proper actions are submanifolds)

Let XX be a smooth manifold, GG a Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

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

(see also this MO discussion)

Proof

Let xX GXx \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 xx in XX is GG-equivariantly diffeomorphic to a linear representation VRO(G)V \in RO(G). The fixed locus V GVV^G \subset V of that is hence diffeomorphic to an open neighbourhood of xx in Σ\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 GG a topological group, consider the category of TopologicalGSpaces.

For HGH \subset G any subgroup, consider

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

  • the Weyl group W H(G)N G(H)/HTopologicalGroupsW_H(G) \coloneqq N_G(H)/H \in TopologicalGroups.

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

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

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

G/H×N(H)/H G/H (gH,nH) gHnH=gnn 1HnHH=gnH \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:

TopologicalN(H)/HSpaces G/H× N(H)/H() TopologicalGSpaces. \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 HH-fixed locus functor (1) being the right adjoint:

(2)TopologicalGSpaces() HG/H× N(H)/H()TopologicalN(H)/HSpaces. 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 XTopologicalN(H)/HSpacesX \in Topological N(H)/H Spaces and YTopologicalGSpacesY \in Topological G Spaces a GG-equivariant continuous function

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

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

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

Since YY is a fixed locus for HN(H)H \subset N(H), by equivariance this restriction has to factor through the HH-fixed locus X HX^H of XX:

Y G/H× N(H)/HY f˜ f X H 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× N(H)/HYG/H \times_{N(H)/H} Y is an image under the GG-action of a point in YG/H× N(H)/HY \subset G/H \times_{N(H)/H}, this restriction f˜\tilde f already determines ff uniquely.

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

Remark

The adjunction (2) factors as:

TopologicalGSpacesG× N(H)()TopologicalN(H)Spaces() HN(H)/H× N(H)/H()TopologicalN(H)/HSpaces. 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× N(H)/H()N(H)/H \times_{N(H)/H} (-), is the identity on the underlying topological spaces, but extends the action from N(H)/HN(H)/H to N(H)N(H), namely through the projection homomorphims N(H)N(H)/HN(H) \to N(H)/H.

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


Last revised on April 8, 2021 at 01:28:28. See the history of this page for a list of all contributions to it.