# 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 20:38, 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.