Oriented Bivariant Theories, I

Abstract

In 1981 W. Fulton and R. MacPherson introduced the notion of bivariant theory (BT), which is a sophisticated unification of covariant theories and contravariant theories. This is for the study of singular spaces. In 2001 M. Levine and F. Morel introduced the notion of algebraic cobordism, which is a universal oriented Borel–Moore functor with products (OBMF) of geometric type, in an attempt to understand better V. Voevodsky's (higher) algebraic cobordism. In this paper we introduce a notion of oriented bivariant theory (OBT), a special case of which is nothing but the oriented Borel–Moore functor with products. The present paper is a first one of the series to try to understand Levine–Morel's algebraic cobordism from a bivariant theoretical viewpoint, and its first step is to introduce OBT as a unification of BT and OBMF.

Motivic Bivariant Characteristic Classes and Related Topics

“…As an "oriented" analogue of the pre-motivic bivariant theory M( [44] we showed the following counterpart of Theorem 3.1, with (more or less) the same definition of the bivariant operations and the first Chern class operator similarly to (1) (given right after Definition-Theorem 6.2):…”

“…As the "confined" and "specialized" maps we take the class Prop of proper and Sm of smooth morphisms, respectively. Theorem 3.1 ( [44], [38]). We define…”

“…In [44] we introduced (in greater generality) the following notion of an oriented bivariant theory. Definition 6.6.…”

“…In this paper we make a survey on the above results [38] as well as on a corresponding universal "oriented" bivariant theory [44], which is a first step on the way to a bivariant-theoretic analogue of Levine-Morel's or Levine-Pandharipande's algebraic cobordism [28,29]. Finally we switch to a differential topological context of smooth manifolds and make a remark on a new geometric bivariant bordism theory based on the notion of a "fiberwise bordism" ( [3], [45]).…”

Motivic bivariant characteristic classes

“…The bivariant theory M(V/−) has the following universal property (see [41,Theorem 3.1] for the proof of a more general result): satisfying the normalization condition that for a smooth morphism f : X → Y the follow-…”