MOTIVIC BIVARIANT CHARACTERISTIC CLASSES AND RELATED TOPICS

We have recently constructed a bivariant analogue of the motivic Hirzebruch classes. A key idea is the construction of a suitable universal bivariant theory in the algebraic-geometric (or compact complex analytic) context, together with a corresponding "bivariant blow-up relation" generalizing Bittner's presentation of the Grothendieck group of varieties. Before we already introduced a corresponding universal "oriented" bivariant theory as an intermediate step on the way to a bivariant analogue of Levine-Morel's algebraic cobordism. Switching to the differential topological context of smooth manifolds, we similarly get a new geometric bivariant bordism theory based on the notion of a "fiberwise bordism". In this paper we make a survey on these theories.