Assignments for topological group actions

TL;DR

This paper studies polynomial assignments for compact torus actions on topological spaces, establishing their algebraic properties and relations to equivariant cohomology, with applications to Hamiltonian symplectic manifolds.

Contribution

It introduces and analyzes the algebra of polynomial assignments, proving localization, a Chang-Skjelbred lemma, and a GKM presentation, with new surjectivity criteria for Hamiltonian actions.

Findings

Established Borel localization for polynomial assignments

Proved a Chang-Skjelbred type lemma for the assignment algebra

Provided a GKM-type presentation and surjectivity criteria for Hamiltonian actions

Abstract

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.