Bivariant Versions of Algebraic Cobordism

We define four distinct oriented bivariant theories associated with algebraic cobordism in its two versions (the axiomatic $\mathrm{\Omega }$ and the geometric $\omega$), when applied to quasi-projective varieties over a field $k$. Specifically, we obtain contravariant analogues of the algebraic bordism group ${\mathrm{\Omega }}_{\ast }(X)$ and the double point bordism group ${\omega }_{\ast }(X)$, for $X$ a quasi-projective variety, and covariant analogues of the algebraic cobordism ring ${\mathrm{\Omega }}^{\ast }(X)$ and the double point cobordism ring ${\omega }^{\ast }(X)$, 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.