  A polynomial assignment for a continuous action of a compact torus $T$ on a
topological space $X$ assigns to each $p\in X$ 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 ${\mathcal{A}}_T(X)$ of all polynomial
assignments has a natural structure of an algebra over the polynomial ring of
${\rm Lie}(T)$. It is an equivariant homotopy invariant, canonically related to
the equivariant cohomology algebra. In this paper we prove various properties
of ${\mathcal{A}}_T(X)$ 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.

Algebraic Topology;Symplectic Geometry