Cobordism bicycles of vector bundles

Abstract

The main ingredient of the algebraic cobordism of M. Levine and F. Morel is a cobordism cycle of the form $(M \xrightarrow {h} X; L_1, \cdots, L_r)$ with a proper map $h$ from a smooth variety $M$ and line bundles $L_i$'s over $M$. In this paper we consider a \emph{cobordism bicycle} of a finite set of line bundles $(X \xleftarrow p V \xrightarrow s Y; L_1, \cdots, L_r)$ with a proper map $p$ and a smooth map $s$ and line bundles $L_i$'s over $V$. We will show that the Grothendieck group $\mathscr Z^*(X, Y)$ of the abelian monoid of the isomorphism classes of cobordism bicycles of finite sets of line bundles satisfies properties similar to those of Fulton-MacPherson's bivariant theory and also that $\mathscr Z^*(X, Y)$ is a \emph{universal} one among such abelian groups, i.e., for any abelian group $\mathscr B^*(X, Y)$ satisfying the same properties there exists a unique Grothendieck transformation $\gamma: \mathscr Z^*(X,Y) \to \mathscr B^*(X,Y)$ preserving the unit.