The cohomology rings of abelian symplectic quotients

TL;DR

This paper provides explicit formulas for the rational cohomology rings of abelian symplectic quotients, linking them to the original manifold's cohomology and fixed point data, with conditions for integral cohomology and torsion-freeness.

Contribution

It introduces a new explicit formula for the cohomology rings of symplectic quotients under torus actions, extending previous results and enabling torsion analysis.

Findings

Explicit formulas for rational cohomology rings of symplectic quotients

Conditions under which integral cohomology formulas apply

Cases where the cohomology is torsion-free

Abstract

Let $M$ be a symplectic manifold, equipped with a Hamiltonian action of a torus $T$. We give an explicit formula for the rational cohomology ring of the symplectic quotient $M//T$ in terms of the cohomology ring of $M$ and fixed point data. Under some restrictions, our formulas apply to integral cohomology. In certain cases these methods enable us to show that the cohomology of the reduced space is torsion-free.