Equivariant cobordism for torus actions

We study the equivariant cobordism groups for the action of a split torus T on varieties over a field k of characteristic zero. We show that for T acting on a variety X. there is an isomorphism Omega(T)(*) (X) circle times(Omega)*(BT) L congruent to(->) Omega(*)(X). As applications, we show that for a connected linear algebraic group G acting on a k-variety X, the forgetful map Omega(G)(*)(X) -> Omega(*)(X) is surjective with rational coefficients. As a consequence, we describe the rational algebraic cobordism ring of algebraic groups and flag varieties. We prove a structure theorem for the equivariant cobordism of smooth projective varieties with torus action. Using this, we prove various localization theorems and a form of Bolt residue formula for such varieties. As an application, we show that the equivariant cobordism of a smooth variety X with torus action is generated by the invariant cobordism cycles in Omega(*)(X) as Omega*(BT)-module. (C) 2012 Elsevier Inc. All rights reserved.