Assignments for topological group actions

A polynomial assignment for a continuous action of a compact torus T on a topological space X assigns to each p∈X a polynomial function on the Lie algebra of the isotropy group at p in such a way that a certain compatibility condition is satisfied. The space AT(X) of all polynomial assignments has a natural structure of an algebra over the polynomial ring of Lie(T). It is an equivariant homotopy invariant, canonically related to the equivariant cohomology algebra. In this paper we prove various properties of AT(X) such as Borel localization, a Chang-Skjelbred lemma, and a Goresky-Kottwitz-MacPherson presentation. In the special case of Hamiltonian torus actions on symplectic manifolds we prove a surjectivity criterion for the assignment equivariant Kirwan map corresponding to a circle in T. We then obtain a Tolman-Weitsman type presentation of the kernel of this map.