Co-operational bivariant theory

Yokura, Shoji

Mathematics Research Reports, Tome 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.

Classification : 55N35, 55S99, 14F99
Keywords: bivariant theory, operational bivariant theory, cohomology operation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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