Bivariant Versions of Algebraic Cobordism

TL;DR

This paper develops four new oriented bivariant theories related to algebraic cobordism, connecting axiomatic and geometric versions, and establishing their properties over fields of characteristic zero.

Contribution

It introduces contravariant and covariant bivariant theories for algebraic and double point cobordism, expanding the framework of algebraic cobordism in a novel way.

Findings

Constructed four distinct oriented bivariant theories.

Established correspondences between theories over characteristic zero fields.

Extended algebraic cobordism concepts to bivariant settings.

Abstract

We define four distinct oriented bivariant theories associated with algebraic cobordism in its two versions (the axiomatic $\Omega$ and the geometric $\omega$), when applied to quasi-projective varieties over a field $k$. Specifically, we obtain contravariant analogues of the algebraic bordism group $\Omega_*(X)$ and the double point bordism group $\omega_*(X)$, for $X$ a quasi-projective variety, and covariant analogues of the algebraic cobordism ring $\Omega^*(X)$ and the double point cobordism ring $\omega^*(X)$, for $X$ a smooth variety. When the ground field has characteristic zero, we use the universal properties of algebraic cobordism in order to obtain correspondences between these oriented bivariant theories.