Motivic Bivariant Characteristic Classes and Related Topics

Abstract

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.

Motivic bivariant characteristic classes

“…in the algebraic geometric context for the construction of a "geometric" bivariant-theoretic version of Levine-Morel's algebraic cobordism (different from the "operational" vivariant theory of [20], e.g. see [41,36]). Similarly, in [2] we will construct in the context of reduced differentiable spaces a "geometric" bivariant-theoretic version of Quillen's complex cobordism (different from the abstract definition given by the general theory of Fulton-MacPherson [19]), and closely related to the approach of Emerson-Meyer [14,15] to "(bivariant) KK-theory via correspondences".…”