Orientation reversal of manifolds

TL;DR

This paper investigates the properties of orientable manifolds regarding their symmetries, proving the existence of strongly chiral manifolds in all dimensions above two and constructing specific examples with particular symmetry constraints.

Contribution

It establishes the existence of strongly chiral manifolds in all dimensions greater than two, including simply-connected examples in dimensions above six, and constructs lens spaces with specific symmetry properties.

Findings

Existence of strongly chiral, smooth manifolds in every oriented bordism class for dimensions >2.

Construction of simply-connected, strongly chiral manifolds in dimensions >6.

Lens spaces with orientation-reversing diffeomorphisms of order 2^k but no degree -1 self-map of smaller order.

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 greater than two. We also produce simply-connected, strongly chiral manifolds in every dimension greater than six. For every positive integer k, we exhibit lens spaces with an orientation-reversing self-diffeomorphism of order 2^k but no self-map of degree -1 of smaller order.