 

Shoji Yokura
Cobordism bicycles of vector bundles
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Published: 
September 9, 2020. 
Keywords: 
(co)bordism, algebraic cobordism, algebraic cobordism of bundles, correspondence. 
Subject: 
55N35, 55N22, 14C17, 14C40, 14F99, 19E99. 


Abstract
The main ingredient of the algebraic cobordism of M. Levine and F. Morel is a cobordism cycle of the form
(h: M → X; L_{1}, ..., 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; L_{1}, ..., 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 FultonMacPherson'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.


Acknowledgements
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.


Author information
Shoji Yokura:
Graduate School of Science and Engineering
Kagoshima University
2135 Korimoto 1chome, Kagoshima 8900065, Japan
yokura@sci.kagoshimau.ac.jp

