Beyond cohomological assignments

Abstract

Let a torus T act in a Hamiltonian fashion on a compact symplectic manifold $(M,\omega )$. The assignment ring ${\mathcal{A}}_T(M)$ is an extension of the equivariant cohomology ring $H_T(M)$; it is modeled on the GKM description of the equivariant cohomology of a GKM space. We show that ${\mathcal{A}}_T(M)$ is a finitely generated $\mathbb{S}({\mathfrak{t}}^{⁎})$-module, and give a criterion guaranteeing that a given set of assignments generates (alternatively, is a basis for) this module. We define two new types of assignments, delta classes and bridge classes, and show that if the torus T is 2-dimensional, then all assignments of sufficiently high degree are generated by cohomological, delta, and bridge classes. In particular, if M is 6-dimensional, then we can find a basis of such classes.