Co-operational bivariant theory
type: Item
instance of: scholarly article
title: Co-operational bivariant theory
author: Shoji Yokura
published in: Mathematics Research Reports
publication date: 2024-08-12T00:00:00Z
review text: Let \(\mathcal{C}\) be a category that has a final object \(pt\) and in which the fiber product is well-defined. Additionally, classes of ``confined maps'' and ``independent squares'' are distinguished. The main example is the category of complex algebraic varieties (or just pseudomanifolds), with the distinguished classes to be the proper maps and pull-back squares. The concept of bivariant theory was introduced in [\textit{W. Fulton} and \textit{R. MacPherson}, Categorical framework for the study of singular spaces. Providence, RI: American Mathematical Society (AMS) (1981; Zbl 0467.55005)], see also [\textit{S. Yokura}, Topology Appl. 123, No. 2, 283--296 (2002; Zbl 1045.55003)]. Every covariant functor, such as a homology theory \(h_*(-)\), gives rise to its bivariant counterpart: for a map \(X \to Y\), a graded abelian group \(\mathbb{B}^{op}h_*(X \to Y)\) is assigned. A dual notion, called ``co-operational'' bivariant theory, is defined axiomatically. Any cohomology theory defines such a co-operational theory. A transformation of multiplicative cohomology theories induces a transformation of co-operational bivariant theories. In particular, if \(h^*(-)=H^*(-)\) is the standard cohomology, then the co-operational theory contains information about both: cohomlogy \(H^*(X) = \mathbb{B}^{coop}H^*(X \to X)\) and the Borel-Moore homology, which is isomorphic to \(\mathbb{B}^{coop}H^*(X \to pt)\). Examples of theories such \(K\)-theory, Chow groups, Levine-Morel cobordisms and transformations: the Chern character, Adams operations are discussed.
reviewed by: Andrzej Weber
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 55S99
Mathematics Subject Classification ID: 14F99
zbMATH DE Number: 7895132
zbMATH Keywords: bivariant theory
zbMATH Keywords: operational bivariant theory
zbMATH Keywords: cohomology operation
MaRDI profile type: Publication
cites work: Equivariant Grothendieck-Riemann-Roch and localization in operational \(K\)-theory. With an appendix by Gabriele Vezzosi.
cites work: Operational \(K\)-theory
cites work: On Grothendieck transformations in Fulton-MacPherson's bivariant theory
cites work: Steenrod operations in Chow theory
cites work: Lectures on algebraic topology.
cites work: Q3328651
cites work: Categorical framework for the study of singular spaces
cites work: Bivariant algebraic cobordism
cites work: La théorie des classes de Chern
cites work: Q3858946
cites work: \(K\)-theory. An introduction. With a new postface by the author and a list of errata
cites work: Multisingularities, cobordisms, and enumerative geometry
cites work: Steenrod operations, degree formulas and algebraic cobordism
cites work: Algebraic Cobordism
cites work: Algebraic cobordism revisited
cites work: Steenrod operations and degree formulas
cites work: Mixed Hodge Structures
cites work: Elementary proofs of some results of cobordism theory using Steenrod operations
cites work: On the structure of algebraic cobordism
cites work: THE MULTIPLICATIVE GROUP OF CLASSICAL COHOMOLOGY
cites work: Cohomology operations, and obstructions to extending continuous functions
cites work: Cohomology Operations (AM-50)
cites work: The total Chern class is not a map of multiplicative cohomology theories
cites work: Stable and unstable operations in algebraic cobordism
cites work: Q3559373
cites work: Bivariant theories of constructible functions and Grothendieck transformations
cites work: Localized operations on \(T\)-equivariant oriented cohomology of projective homogeneous varieties
DOI: 10.5802/MRR.20
description: scientific article; zbMATH DE number 7895132