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–MacPherson’s bivariant theory $\mathbb{B}(X\stackrel{f}{\to }Y)$ defined for a morphism $f:X\to Y$. In this paper, using correspondences we construct a bi-variant algebraic cobordism ${\mathrm{\Omega }}^{\ast ,\mathrm{♯}}(X,Y)$ such that ${\mathrm{\Omega }}^{\ast ,\mathrm{♯}}(X,pt)$ is isomorphic to Lee–Pandharipande’s algebraic cobordism of vector bundles ${\mathrm{\Omega }}_{-\ast ,\mathrm{♯}}(X)$. In particular, ${\mathrm{\Omega }}^{\ast }(X,pt)={\mathrm{\Omega }}^{\ast ,0}(X,pt)$ is isomorphic to Levine–Morel’s algebraic cobordism ${\mathrm{\Omega }}_{-\ast }(X)$. Namely, ${\mathrm{\Omega }}^{\ast ,\mathrm{♯}}(X,Y)$ is \emph{a bi-variant vesion} of Lee–Pandharipande’s algebraic cobordism of bundles ${\mathrm{\Omega }}_{\ast ,\mathrm{♯}}(X)$.