Chiral manifold

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A closed, connected, orientable manifold in one of the categories [[Wikipedia:Topological_manifold|TOP]], [[Wikipedia:Piecewise_linear_manifold|PL]] or [[Wikipedia:Smooth_manifold|DIFF]] is called ''chiral'' if it does not admit an orientation-reversing automorphism in the respective category and ''amphicheiral'' if it does. For the sake of clarity, the category should be indicated by adverbs: e. g. a ''topologically chiral'' manifold does not admit an orientation-reversing self-homeomorphism, whereas a ''smoothly amphicheiral'' manifold is a differentiable manifold which admits an orientation-reversing self-diffeomorphism.
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A closed, connected, orientable manifold in one of the categories [[Wikipedia:Topological_manifold|TOP]], [[Wikipedia:Piecewise_linear_manifold|PL]] or [[Wikipedia:Smooth_manifold|DIFF]] is called ''chiral'' if it does not admit an orientation-reversing automorphism in the respective category and ''amphicheiral'' if it does. For the sake of clarity, the category should be indicated by adverbs: e. g. a ''topologically chiral'' manifold does not admit an orientation-reversing self-homeomorphism, whereas a ''smoothly amphicheiral'' manifold is a differentiable manifold which admits an orientation-reversing self-diffeomorphism.
This definition can be extended by the notion of ''homotopical chirality/amphicheirality'' when homotopy self-equivalences are considered. Chiral manifolds in the strongest sense do not admit self-maps of degree −1; they are called ''strongly chiral'' and ''weakly amphicheiral'' in the opposite case.
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This definition can be extended by the notion of ''homotopical chirality/amphicheirality'' when homotopy self-equivalences are considered. Chiral manifolds in the strongest sense do not admit self-maps of degree −1; they are called ''strongly chiral'' and ''weakly amphicheiral'' in the opposite case.
== Terminology ==
== Terminology ==
The terminology ''amphicheiral'' was introduced by Tait {{cite|Tait1876}} (p.160) in his work on knots.
The terminology ''amphicheiral'' was introduced by Tait {{cite|Tait1876}} (p.160) in his work on knots.
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== References ==
== References ==
* {{Müllner2009}}
* {{Müllner2009}}
* {{Tait1876}}
* {{Tait1876}}
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== External links ==
[[Category:Theory]] [[Category:Aspherical]]
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* The Wikipedia page about [[Wikipedia:Chirality_(mathematics)|chirality in mathematics]].
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[[Category:Definitions]]

Latest revision as of 10:48, 13 June 2013

A closed, connected, orientable manifold in one of the categories TOP, PL or DIFF is called chiral if it does not admit an orientation-reversing automorphism in the respective category and amphicheiral if it does. For the sake of clarity, the category should be indicated by adverbs: e. g. a topologically chiral manifold does not admit an orientation-reversing self-homeomorphism, whereas a smoothly amphicheiral manifold is a differentiable manifold which admits an orientation-reversing self-diffeomorphism.

This definition can be extended by the notion of homotopical chirality/amphicheirality when homotopy self-equivalences are considered. Chiral manifolds in the strongest sense do not admit self-maps of degree −1; they are called strongly chiral and weakly amphicheiral in the opposite case.

[edit] 1 Terminology

The terminology amphicheiral was introduced by Tait [Tait1876] (p.160) in his work on knots.

The words amphicheiral, amphichiral and achiral are synonyms. Amphicheiral is most frequently used in MathSciNet.

[edit] 2 References

  • P.G.Tait, On knots I., Trans. Roy. Soc. Edin. 28 (1876), 145–190.

[edit] 3 External links

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