The main ingredient of the algebraic cobordism of M. Levine and F. Morel is a cobordism cycle of the form (h: M → X; L₁, ..., 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 cobordism bicycle of a finite set of line bundles (p: V → X, s: V → Y; L₁, ..., 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 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 Z*(X, Y) is a universal one among such abelian groups, i.e., for any abelian group B*(X, Y) satisfying the same properties there exists a unique Grothendieck transformation γ: Z*(X,Y) → B*(X,Y) preserving the unit.

The author would like to thank the anonymous referee for his/her careful reading of the manuscript, valuable comments, and constructive suggestions. This work was supported by JSPS KAKENHI Grant Numbers JP16H03936 and JP19K03468.