Involutions in lens spaces

Lens spaces arise from free periodic maps on odd-dimensional spheres. They admit free involutions. We show that if the period of the map is odd, the mod two invariants of such involutions are very much like those of the antipodal map on the sphere, both for single lens spaces and for bundles of lens spaces. In particular, Borsuk–Ulam type results over a base space can be obtained for such lens spaces, just as for the spheres. On the other hand, the lens spaces arising from periodic maps of even periods behave differently. For instance, there exists an equivariant map from the three-dimensional real projective space to the 2-sphere.