A bi-variant algebraic cobordism via correspondences

A bi-variant theory $\mathbb{B}(X,Y)$ defined for a pair $(X,Y)$ is a theory satisfying properties similar to those of Fulton–Mac Pherson’s bivariant theory $\mathbb{B}(X \xrightarrow{f} Y)$ defined for a morphism $f : X \rightarrow Y$. In this paper, using correspondences we construct a bi-variant algebraic cobordism $\Omega^{\ast,\sharp} (X, Y )$ such that $\Omega^{\ast,\sharp}(X, pt)$ is isomorphic to Lee–Pandharipande’s algebraic cobordism of vector bundles $\Omega \underline{}_{\ast,\sharp} (X)$. In particular, $\Omega^\ast (X, pt) = \Omega^{\ast,0} (X, pt)$ is isomorphic to Levine–Morel’s algebraic cobordism $\Omega \underline{}_{\ast} (X)$. Namely, $\Omega^{\ast,\sharp} (X,Y)$ is a bi-variant version of Lee–Pandharipande’s algebraic cobordism of bundles $\Omega_{\ast,\sharp} (X)$.