Moment Maps and Non-compact Cobordisms

We define a moment map for a torus action on a smooth manifold without a two-form. Cobordisms of such structures are meaningful even if the manifolds are non compact, as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant the normal bundle of its fixed point set. Two formulas follow easily: Guillemin’s topological version of the abelian Jeffrey-Kirwan localization, and the Guillemin-Lerman-Sternberg formula for the Duistermaat-Heckman measure.