fixed point space in nLab

Contents

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:

• fixed point spectra,

• geometric fixed point spectra

• categorical fixed point spectra?

In equivariant differential topology

In equivariant differential topology:

(Kankaanrinta 07, theorem 4.4, theorem 4.6)

(see also this MO discussion)

Properties

Fixed point adjunction

For GG a topological group, consider the category of TopologicalGSpaces.

For H⊂GH \subset G any subgroup, consider

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

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

Observing that for X∈TopologicalGSpacesX \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 −1Hn⏟HH=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 } \,.

• fixed point group

• homotopy fixed points

• Elmendorf's theorem

• equivariant homotopy theory

• equivariant differential topology