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Co-operational bivariant theory
Shoji Yokura1
1 Graduate School of Science and Engineering, Kagoshima University, 1-21-35 Korimoto, Kagoshima, 890-0065, Japan
Mathematics Research Reports, Volume 5 (2024), pp. 21-55.

For a covariant functor W. Fulton and R. MacPherson defined an operational bivariant theory associated to this covariant functor. In this paper we will show that given a contravariant functor one can similarly construct a “dual” version of an operational bivariant theory, which we call a co-operational bivariant theory. If a given contravariant functor is the usual cohomology theory, then our co-operational bivariant group for the identity map consists of what are usually called “cohomology operations”. In this sense, our co-operational bivariant theory consists of “generalized” cohomology operations.

Received:
Revised:
Published online:
DOI: 10.5802/mrr.20
Classification: 55N35, 55S99, 14F99
Keywords: bivariant theory, operational bivariant theory, cohomology operation
Shoji Yokura 1

1 Graduate School of Science and Engineering, Kagoshima University, 1-21-35 Korimoto, Kagoshima, 890-0065, Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Shoji Yokura. Co-operational bivariant theory. Mathematics Research Reports, Volume 5 (2024), pp. 21-55. doi : 10.5802/mrr.20. https://mrr.centre-mersenne.org/articles/10.5802/mrr.20/

[1] Dave Anderson; Richard Gonzales; Sam Payne Equivariant Grothendieck–Riemann–Roch and localization in operational K-theory, Algebra Number Theory, Volume 15 (2021) no. 2, pp. 341-385 | DOI | MR | Zbl

[2] Dave Anderson; Sam Payne Operational K-theory, Doc. Math., Volume 20 (2015), pp. 357-399 | DOI | MR | Zbl

[3] Jean-Paul Brasselet; Joerg Schürmann; Shoji Yokura On Grothendieck transformations in Fulton–MacPherson’s bivariant theory, J. Pure Appl. Algebra, Volume 211 (2007), pp. 665-684 | DOI | MR | Zbl

[4] Patrick Brosnan Steenrod operations in Chow theory, Trans. Am. Math. Soc., Volume 355 (2003) no. 5, pp. 1869-1903 | DOI | MR | Zbl

[5] Albrecht Dold Lectures on Algebraic Topology, Springer, 1995 | DOI

[6] William Fulton Intersection Theory, Springer, 1984 | DOI

[7] William Fulton; Robert MacPherson Categorical Framework For the Study of Singular Spaces, Memoirs, 243, American Mathematical Society, 1981

[8] José González; Kalle Karu Bivariant algebraic cobordism, Algebra Number Theory, Volume 9 (2015) no. 6, pp. 1293-1336 | DOI | MR | Zbl

[9] Alexander Grothendieck La théorie des classes de Chern, Bull. Soc. Math. Fr., Volume 86 (1958), pp. 137-154 | DOI | Numdam | Zbl

[10] Robert M. Hardt; Clint G. McCrory Steenrod operations in subanalytic homology, Compos. Math., Volume 39 (1979) no. 3, pp. 333-371 | Numdam | MR | Zbl

[11] Max Karoubi K-Theory; An introduction, Classics in Mathematics, 226, Springer, 1978 | DOI

[12] Maxim E. Kazarian Multisingularities, cobordisms, and enumerative geometry, Russ. Math. Surv., Volume 58 (2003) no. 4, pp. 665-724 | DOI | Zbl

[13] Marc Levine Steenrod operations, degree formulas and algebraic cobordism, Pure Appl. Math. Q., Volume 3 (2007) no. 1, pp. 283-306 | DOI | MR | Zbl

[14] Marc Levine; Fabien Morel Algebraic Cobordism, Springer Monographs in Mathematics, Springer, 2007

[15] Marc Levine; Rahul Pandharipande Algebraic cobordism revisited, Invent. Math., Volume 176 (2009), pp. 63-130 | DOI | MR | Zbl

[16] Alexander Merkurjev Steenrod operations and degree formulas, J. Reine Angew. Math., Volume 565 (2003), pp. 13-26 | DOI | MR | Zbl

[17] nLab authors Adams operation, https://ncatlab.org/nlab/show/Adams+operation, 2024 (Revision 27)

[18] nLab authors cohomology operation, https://ncatlab.org/nlab/show/cohomology+operation, 2024 (Revision 32)

[19] Toru Ohmoto Multi-singularities and Hilbert’s 15th problem (2023) Lecture Notes (in Japanese) for the conference “Singularities of Differentiable Maps and Its Applications—In commemoration of Goo Ishikawa’s all years of hard work”, at Hokkaido University, March 6-8

[20] Chris A. M. Peters; Joseph H. M. Steenbrink Mixed Hodge Structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 52, Springer, 2008 | DOI

[21] Daniel Quillen Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. Math., Volume 7 (1971) no. 1, pp. 29-56 | DOI | MR | Zbl

[22] Pavel Sechin On the structure of algebraic cobordism, Adv. Math., Volume 333 (2018), pp. 314-349 | DOI | MR | Zbl

[23] Graeme Segal The multiplicative group of classical cohomology, Q. J. Math. Oxf., Volume 26 (1975) no. 1, pp. 289-293 | DOI | MR | Zbl

[24] Norman E. Steenrod Cohomology Operations and Obstructions to Extending Continuous Functions, 1957 https://faculty.tcu.edu/gfriedman/notes/steenrod.pdf (Colloquium Lectures, the Annual Summer Meeting of the Amer. Math. Soc. at Pennsylvania State University)

[25] Norman E. Steenrod Cohomology Operations and Obstructions to Extending Continuous Functions, Adv. Math., Volume 8 (1972) no. 3, pp. 371-416 | DOI | MR | Zbl

[26] Norman E. Steenrod; David B. A. Epstein Cohomology Operations, Annals of Mathematics Studies, 50, Princeton University Press, 1962 https://people.math.rochester.edu/faculty/doug/otherpapers/steenrod-epstein.pdf

[27] Burt Totaro The total Chern class is not a map of multiplicative cohomology theories, Math. Z., Volume 212 (1993), pp. 527-532 | DOI | MR | Zbl

[28] Alexander Vishik Algebraic cobordism, Landwever–Novikov and Steenrod operations, symmetric operations, 2007 (School and Conference on Algebraic K-theory and its Applications, The Abdus Salam ICTP)

[29] Alexander Vishik Stable and Unstable operations in Algebraic Cobordism, Ann. Sci. Éc. Norm. Supér., Volume 52 (2019) no. 3, pp. 561-630 | DOI | MR | Zbl

[30] Shoji Yokura Co-operational bivariant theory, II (in preparation)

[31] Shoji Yokura Bivariant theories of constructible functions and Grothendieck transformations, Topology Appl., Volume 123 (2002) no. 2, pp. 283-296 | DOI | MR | Zbl

[32] Kirill Zainoulline Localized operations on T-equivariant oriented cohomology of projective homogeneous varieties, J. Pure Appl. Algebra, Volume 226 (2022) no. 3, 106856 | DOI | MR | Zbl

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