A bi-variant theory $\mathbb B(X,Y)$ defined for a pair $(X,Y)$ is a theory satisfying properties similar to those of Fulton--MacPherson's bivariant theory $\mathbb B(X \xrightarrow f Y)$ defined for a morphism $f:X \to Y$. In this paper, using correspondences we construct a bi-variant algebraic cobordism $\Omega^{ *,\sharp}(X, Y)$ such that $\Omega^{ *,\sharp}(X, pt)$ is isomorphic to Lee--Pandharipande's algebraic cobordism of vector bundles $\Omega_{ -*,\sharp}(X)$. In particular, $\Omega^{ *}(X, pt)=\Omega^{ *, 0}(X, pt)$ is isomorphic to Levine--Morel's algebraic cobordism $\Omega_{ -*}(X)$. Namely, $\Omega^{ *,\sharp}(X, Y)$ is \emph{ a bi-variant vesion} of Lee--Pandharipande's algebraic cobordism of bundles $\Omega_{ *,\sharp}(X)$.
Categories: math.AG math.AT math.CT