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

any comments are welcome. Substantial revised version to appear in Pure and Applied Mathematics Quarterly