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. 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. | + | A closed, connected, orientable manifold in one of the categories [[Wikipegia:Topological_manifold|TOP]], [[Piecewise_linear_manifold|PL]] or [[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. | 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. |
Revision as of 21:38, 21 November 2009
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.
(The words amphicheiral, amphichiral and achiral are synonyms. Amphicheiral is most frequently used in MathSciNet.)
References
- Daniel Müllner, Orientation reversal of manifolds, Algebr. Geom. Topol. 9 (2009), no. 4, 2361–2390.