CANONICAL BASES FOR THE EQUIVARIANT COHOMOLOGY AND K-THEORY RINGS OF SYMPLECTIC TORIC MANIFOLDS

Let M be a symplectic toric manifold acted on by a torus T. In this work we exhibit an explicit basis for the equivariant K-theory ring KT(M) which is canonically associated to a generic component of the moment map. We provide a combinatorial algorithm for computing the restrictions of the elements of this basis to the fixed point set; these, in turn, determine the ring structure of KT(M). The construction is based on the notion of local index at a fixed point, similar to that introduced by Guillemin and Kogan in (GK). We apply the same techniques to exhibit an explicit basis for the equivari- ant cohomology ring HT(M;Z) which is canonically associated to a generic component of the moment map. Moreover we prove that the elements of this basis coincide with some well-known sets of classes: the equivariant Poincare duals to the closures of unstable manifolds, and also the canonical classes in- troduced by Goldin and Tolman in (GT), which exist whenever the moment map is index increasing.