Chiral manifold
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== Terminology == | == Terminology == | ||
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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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== External links == | == External links == | ||
* The Wikipedia page about [[Wikipedia:Chirality_(mathematics)|chirality in mathematics]]. | * The Wikipedia page about [[Wikipedia:Chirality_(mathematics)|chirality in mathematics]]. | ||
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[[Category:Definitions]] | [[Category:Definitions]] |
Latest revision as of 11:48, 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.