Orientation reversal of manifolds

Abstract

We study the phenomenon of chirality in the context of manifolds. An orientable manifold is called amphicheiral if it admits an orientation-reversing self-map and chiral if it does not. In addition to being continuous and of degree –1, the self map can be required to be a homotopy equivalence, a homeomorphism or a diffeomorphism. We call a manifold e.g. “smoothly amphicheiral” or “homotopically chiral”, according to the category which is considered.<br /> We show that in every dimension greater or equal to 3, there exist manifolds which are chiral in the strongest sense, i.e. which do not admit a self-map of degree –1. Further, we prove that there are simply-connected manifolds without a self-map of degree –1 in every dimension greater or equal to 7. Further results show the existence of homotopically chiral manifolds in every bordism class in dimensions greater or equal to 3. Along with these existence results, we also study the topological obstructions which can prevent orientation reversal. Aiming in the opposite direction, we show that certain products of 3-dimensional lens spaces are smoothly amphicheiral. For every integer k, we exhibit lens spaces whose orientation can be reversed by a diffeomorphism of finite order 2^k but not by any continuous map of smaller order.