  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.
