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.

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.)


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