Advances in Mathematics ( IF 1.494 ) Pub Date : 2020-01-10 , DOI: 10.1016/j.aim.2020.106976
Victor Guillemin; Susan Tolman; Catalin Zara

Let a torus T act in a Hamiltonian fashion on a compact symplectic manifold $\left(M,\omega \right)$. The assignment ring ${\mathcal{A}}_{T}\left(M\right)$ is an extension of the equivariant cohomology ring ${H}_{T}\left(M\right)$; it is modeled on the GKM description of the equivariant cohomology of a GKM space. We show that ${\mathcal{A}}_{T}\left(M\right)$ is a finitely generated $\mathbb{S}\left({\mathfrak{t}}^{⁎}\right)$-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.

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