# Chiral manifold

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.

## 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.

## 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.

## 3 External links

- The Wikipedia page about chirality in mathematics.