# nLab equivariant Tietze extension theorem

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

A generalization of the Tietze extension theorem to equivariant functions provides conditions under which a continuous and equivariant function from a subspace of a topological G-space to another topological G-space has an extension to a continuous and equivariant function to the full $G$-space.

## Statement

###### Theorem

(Tietze-Gleason extension theorem)

Let

If

or

then $f$ has an extension to an equivariant continuous function $\widehat f$ on all of $X$.

$\array{ A &\overset{f}{\longrightarrow}& E \\ \cap & \nearrow_{\mathrlap{ \widehat{f} }} \\ X }$

Other/more general conditions for the equivariant extension to exist:

###### Proposition

(Jaworowski-extension theorem)

If

1. the ambient domain G-space $X$ is a

2. the domain $A \subset X$ is a

3. the codomain G-space is a

1. locally comact

2. separable metric space

3. such that for every $G$-orbit type $(H)$ in the complement $X \setminus A$

the fixed locus $E^H$ is an absolute neighbourhood retract.

Then every continuous function $f \colon A \to E$ has an extension to a $G$-equivariant continuous function $\widehat f$ on an open neighbourhood $A \subset O_A \subset X$

$\left( \underset{ G/H \subset X \setminus A }{\forall} \; E^H \;\; \text{is absolute neighbourhood retract} \right) \;\;\;\; \Rightarrow \;\;\;\; \left( \underset{\widehat f}{\exists} \;\;\;\; \array{ A &\overset{f}{\longrightarrow}& E \\ \cap & \nearrow_{\mathrlap{ \widehat{f} }} \\ O_A \\ \cap \\ X } \right) \,.$

Moreover, if the above fixed loci $E^H$ are even absolute retracts, then an extension $\widehat f$ exists on all of $X$:

$\left( \underset{ G/H \subset X \setminus A }{\forall} \; E^H \;\; \text{is absolute retract} \right) \;\;\;\; \Rightarrow \;\;\;\; \left( \underset{\widehat f}{\exists} \;\;\;\; \array{ A &\overset{f}{\longrightarrow}& E \\ \cap & \nearrow_{\mathrlap{ \widehat{f} }} \\ X } \right) \,.$