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 $(M\stackrel{h}{\to }X;L_1,\cdots ,L_r)$ 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 $(X\stackrel{p}{\leftarrow }V\stackrel{s}{\to }Y;L_1,\cdots ,L_r)$ 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 ${\mathcal{Z}}^{\ast }(X,Y)$ 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 ${\mathcal{Z}}^{\ast }(X,Y)$ is a \emph{universal} one among such abelian groups, i.e., for any abelian group ${\mathcal{B}}^{\ast }(X,Y)$ satisfying the same properties there exists a unique Grothendieck transformation $\gamma :{\mathcal{Z}}^{\ast }(X,Y)\to {\mathcal{B}}^{\ast }(X,Y)$ preserving the unit.

any comments are welcome; to appear in New York Journal of Mathematics

References in corpus (6)

• Algebraic cobordism
• Bivariant Algebraic Cobordism
• Derived algebraic cobordism
• Precobordism and cobordism
• Algebraic cobordism of filtered vector bundles on varieties: Notes on a work of Lee and Pandharipande
• Enriched categories of correspondences and characteristic classes of singular varieties

Cited by in corpus (1)

• Enriched categories of correspondences and characteristic classes of singular varieties