Free Involutions and Circle Actions in Lens Spaces
by
Jan Jaworowski
Indiana University, Bloomington, Indiana

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 can be obtained for lens spaces (or for bundles of lens spaces) just as for 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.

We also study free actions of the circle group in lens spaces. Again we show that such actions bear a certain resemblance to the standard circle actions in odd dimensional spheres, including the parametrized case. In particular, the S¹-indices of such actions are the same as those of the corresponding circle actions in the spheres. Unlike for involutions, the parity of the period for circle actions is not an issue, so our results apply, in particular, to real projective spaces. We note, however, that the constructions of in this paper have no analogue in the case of S³-actions in lens spaces of dimensions of the form 4n+3.

Date received: June 22, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabc-40.