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equivariant Tietze extension theorem

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Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

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Extra stuff, structure, properties

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topological homotopy theory

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homotopy theory, (∞,1)-category theory, homotopy type theory

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see also algebraic topology

Introductions

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Homotopy groups

Basic facts

Theorems

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 GG-space.

Statement

Theorem

(Tietze-Gleason extension theorem)

Let

If

or

then ff has an extension to an equivariant continuous function f^\widehat f on all of XX.

A f E f^ 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 XX is a

    1. locally comact

    2. separable metric space

    3. of finite dimension

    4. with a finite number of orbit types;

  2. the domain AXA \subset X is a

  3. the codomain G-space is a

    1. locally comact

    2. separable metric space

    3. such that for every GG-orbit type (H)(H) in the complement XAX \setminus A

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

Then every continuous function f:AEf \colon A \to E has an extension to a GG-equivariant continuous function f^\widehat f on an open neighbourhood AO AXA \subset O_A \subset X

(G/HXAE His absolute neighbourhood retract)(f^A f E f^ O A 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 HE^H are even absolute retracts, then an extension f^\widehat f exists on all of XX:

(G/HXAE His absolute retract)(f^A f E f^ 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)

extension theoremscontinuous functionssmooth functions
plain functionsTietze extension theoremWhitney extension theorem
equivariant functionsequivariant Tietze extension theorem

References

Last revised on September 22, 2021 at 11:45:23. See the history of this page for a list of all contributions to it.