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 $G$-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.

### 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:

### 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.

###### 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.

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 $\Sigma$.