nLab equivariant Tietze extension theorem

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

• $G$ be a compact Lie group,

• $X$ a topological G-space,

• $A \subset X$ a sub G-space of $X$,

• $E$ be a finite-dimensional orthogonal $G$-linear representation, regarded as a Euclidean topological G-space,

• $A \overset{f}{\longrightarrow} E$ a $G$-equivariant continuous function.

If

• $A \subset X$ is a compact subspace

or

• $X$ is a normal topological space and $A \subset X$ is a closed subspace

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

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

(Gleason 50, see Palais 60, Theorem 1.4.3)

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

Proposition

(Jaworowski-extension theorem)

If

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

   1. locally comact

   2. separable metric space

   3. of finite dimension

   4. with a finite number of orbit types;

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

   • closed

   • sub-G-space;

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 $X$:

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

(Jaworowski 76, Lashof 81)

Related concepts

extension theorems	continuous functions	smooth functions
plain functions	Tietze extension theorem	Whitney extension theorem
equivariant functions	equivariant Tietze extension theorem

• equivariant principal bundle

• slice theorem

References

• Andrew Gleason, Spaces with a compact Lie group of transformations, Proc. Amer. Math. Soc. 1 (1950), 35-43 (doi:10.1090/S0002-9939-1950-0033830-7)

• Richard Palais, Theorem 1.4.3 in: The classification of $G$-spaces, Memoirs of the American Mathematical Society, Number 36, 1960 (ISBN:978-0-8218-9979-3 pdf, pdf)

• Jan Jaworowski, Equivariant extensions of maps, Pacific J. Math. Volume 45, Number 1 (1973), 229-244 (euclid:1102947720)

• Jan Jaworowski, Extending equivariant maps for compact Lie group actions, Bull. Amer. Math. Soc. Volume 79, Number 4 (1973), 698-701 (euclid:1183534741)

• Jan Jaworowski, Extensions of $G$-maps and Euclidean $G$-retracts, Math Z (1976) 146: 143 (doi:10.1007/BF01187702)

• Jan Jaworowski, An equivariant extension theorem and $G$-retracts with a finite structure, Manuscripta Math (1981) 35: 323 (doi:10.1007/BF01263266)

• Richard Lashof, The Equivariant Extension Theorem, Proceedings of the American Mathematical Society Vol. 83, No. 1 (Sep., 1981), pp. 138-140 (jstor:2043909)