On double points of immersed surfaces

Lens spaces arise from free periodic maps in odd-dimensional spheres. They admit free actions of the circle group. We show that such actions bear a certain resemblance to the standard circle actions in odd-dimensional spheres. In particular, the S1-indices of such actions are the same as those of the corresponding S1-actions in the spheres. 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.