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 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.
- 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.
- The Wikipedia page about chirality in mathematics.