RING STRUCTURES OF RATIONAL EQUIVARIANT COHOMOLOGY RINGS AND RING HOMOMORPHISMS BETWEEN THEM

In this paper, we consider a class of connected oriented (with respect to Q) closed G-manifolds with a non-empty finite fixed point set, each of which is G-equivariantly formal, where G is the circle group S-1. Using localization theorem and equivariant index, we give an explicit description of rational equivariant cohomology of such a G-manifold in terms of algebra. This makes it possible to determine the number of equivariant cohomology rings (up to isomorphism) of such 2- and 4-dimensional G-manifolds. Moreover, we obtain an analytic description of the ring homomorphism between equivariant cohomology rings of such two G-manifolds induced by a G-equivariant map, and show a characterization of the ring homomorphism.