A bi-variant algebraic cobordism via correspondences

Abstract

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 $\Omega_{*,\sharp}(X)$.