Oriented bivariant theories, I

TL;DR

This paper introduces oriented bivariant theories (OBT) as a new framework unifying bivariant theories and algebraic cobordism, aiming to deepen understanding of Levine-Morel's algebraic cobordism through a bivariant perspective.

Contribution

It defines OBT as a novel concept that unifies bivariant theories and algebraic cobordism, providing a new approach to study these theories.

Findings

Introduces the concept of oriented bivariant theory (OBT).

Shows OBT unifies bivariant theories and algebraic cobordism.

Lays groundwork for future research on algebraic cobordism via bivariant theories.

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.