2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73787We 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.17 pages, 5 figures, a QuickTime movie visualizing an exotic involution of S^5 is currently available at http://www.ruhr-uni-bochum.de/mathematik8/puttmann, revised version: corrections of typos, minor changes in the presentation, 1 reference addedGeometric TopologyDifferential Geometry57S25 (Primary), 53C22, 57S17, 57R55 (Secondary)text