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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.
(Tietze-Gleason extension theorem)
Let
$G$ be a compact Lie group,
$X$ a topological G-space,
$E$ 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
then $f$ has an extension to an equivariant continuous function $\widehat f$ on all of $X$.
(Gleason 50, see Palais 60, Theorem 1.4.3)
Other/more general conditions for the equivariant extension to exist:
(Jaworowski-extension theorem)
If
the ambient domain G-space $X$ is a
with a finite number of orbit types;
the domain $A \subset X$ is a
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 $A \subset O_A \subset X$
Moreover, if the above fixed loci $E^H$ are even absolute retracts, then an extension $\widehat f$ exists on all of $X$:
extension theorems | continuous functions | smooth functions |
---|---|---|
plain functions | Tietze extension theorem | Whitney extension theorem |
equivariant functions | equivariant Tietze extension theorem |
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)
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