Orientation reversal of manifolds

Abstract

We call a closed, connected, orientable manifold in one of the categories TOP, PL or DIFF chiral if it does not admit an orientation-reversing automorphism and amphicheiral otherwise. Moreover, we call a manifold strongly chiral if it does not admit a self-map of degree $-1$. We prove that there are strongly chiral, smooth manifolds in every oriented bordism class in every dimension $\ge 3$. We also produce simply-connected, strongly chiral manifolds in every dimension $\ge 7$. For every $k\ge 1$, we exhibit lens spaces with an orientation-reversing self-diffeomorphism of order $2^k$ but no self-map of degree $-1$ of smaller order.