Moment maps and non-compact cobordisms

TL;DR

This paper introduces a new framework for moment maps on non-compact manifolds with torus actions, and demonstrates their cobordism properties, leading to simplified proofs of key localization formulas in symplectic geometry.

Contribution

It defines moment maps without two-forms and establishes cobordism relations for non-compact manifolds, providing new proofs of important localization formulas.

Findings

A moment map can be associated with a torus action without a two-form.

Compact manifolds with torus actions are cobordant to unions of normal bundles of fixed points.

New simplified proofs of Jeffrey-Kirwan and Duistermaat-Heckman formulas.

Abstract

We define a moment map associated to a smooth torus action on a smooth manifold, without a two-form. We define cobordisms of such structures, allowing non compact manifolds as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant to the disjoint union of the normal bundles of the connected components of the fixed points set. We use this to give simple new proofs of two formulas: Guillemin's topological version of the abelian Jeffrey-Kirwan localization, and the Guillemin-Lerman-Sternberg formula for the Duistermaat-Heckman measure.