Abstract
We provide explicit, simple, geometric formulas for free involutions ρ of Euclidean spheres that are not conjugate to the antipodal involution. Therefore the quotient Sn/ρ is a manifold that is homotopically equivalent but not diffeomorphic to . We use these formulas for constructing explicit non-trivial elements in π1 Diff(S5) and π1 Diff(S13) and to provide explicit formulas for non-cancellation phenomena in group actions.
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