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Cobordism bicycles of vector bundles. (English) Zbl 1504.55006

Summary: The main ingredient of the algebraic cobordism of M. Levine and F. Morel [Algebraic cobordism. Berlin: Springer (2007; Zbl 1188.14015)] is a cobordism cycle of the form \((M \xrightarrow{h} X; L_1, \dots, L_r)\) 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 \((X \xleftarrow{p} V \xrightarrow{s} Y; L_1, \dots, L_r)\) 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 \(\mathscr{Z}^\ast(X, Y)\) 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 \(\mathscr{Z}^\ast(X, Y)\) is a universal one among such abelian groups, i.e., for any abelian group \(\mathscr{B}^\ast(X, Y)\) satisfying the same properties there exists a unique Grothendieck transformation \(\gamma: \mathscr{Z}^\ast(X, Y) \rightarrow \mathscr{B}^\ast(X, Y)\) preserving the unit.

MSC:

55N35 Other homology theories in algebraic topology
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C40 Riemann-Roch theorems
14F99 (Co)homology theory in algebraic geometry
19E99 \(K\)-theory in geometry

Citations:

Zbl 1188.14015
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References:

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