# Chiral manifold

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− | 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. | + | 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 | + | 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. |

− | (The words ''amphicheiral'', ''amphichiral'' and ''achiral'' are synonyms. ''Amphicheiral'' is most frequently used in [http://www.ams.org/mathscinet/ MathSciNet]. | + | == Terminology == |

+ | The terminology ''amphicheiral'' was introduced by Tait {{cite|Tait1876}} (p.160) in his work on knots. | ||

+ | |||

+ | The words ''amphicheiral'', ''amphichiral'' and ''achiral'' are synonyms. ''Amphicheiral'' is most frequently used in [http://www.ams.org/mathscinet/ MathSciNet]. | ||

== References == | == References == | ||

− | |||

* {{Müllner2009}} | * {{Müllner2009}} | ||

− | + | * {{Tait1876}} | |

− | [[Category: | + | == External links == |

+ | * The Wikipedia page about [[Wikipedia:Chirality_(mathematics)|chirality in mathematics]]. | ||

+ | [[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

- Daniel Müllner,
*Orientation reversal of manifolds*, Algebr. Geom. Topol.**9**(2009), no. 4, 2361–2390.

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

## [edit] 3 External links

- The Wikipedia page about chirality in mathematics.