Wiedersehen metrics and exotic involutions of Euclidean spheres

TL;DR

This paper introduces explicit formulas for certain involutions on Euclidean spheres, leading to new insights into manifold structures, diffeomorphism groups, and group action phenomena.

Contribution

It provides simple geometric formulas for exotic involutions on spheres and applies them to construct non-trivial elements in diffeomorphism groups and study non-cancellation in group actions.

Findings

Explicit formulas for non-antipodal involutions on spheres

Construction of non-trivial elements in pi_1 Diff(S^5) and pi_1 Diff(S^13)

Examples of non-cancellation phenomena in group actions

Abstract

We provide explicit, simple, geometric formulas for free involutions rho of Euclidean spheres that are not conjugate to the antipodal involution. Therefore the quotient S^n/rho is a manifold that is homotopically equivalent but not diffeomorphic to RP^n. We use these formulas for constructing explicit non-trivial elements in pi_1 Diff(S^5) and pi_1 Diff(S^13) and to provide explicit formulas for non-cancellation phenomena in group actions.