Bivariant Versions of Algebraic Cobordism

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.