A More General Method to Classify up to Equivariant $KK$-Equivalence

Using a homological invariant together with an obstruction class in a certain Ext ² -group, we classify objects in triangulated categories that have projective resolutions of length two. This invariant gives strong classification results for actions of the circle group on C^ -algebras, C^ -algebras over finite unique path spaces, and graph C^ -algebras with finitely many ideals.