Torsion algebraic cycles and complex cobordism
HTML articles powered by AMS MathViewer
- by Burt Totaro
- J. Amer. Math. Soc. 10 (1997), 467-493
- DOI: https://doi.org/10.1090/S0894-0347-97-00232-4
- PDF | Request permission
References
- J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0402720
- M. F. Atiyah, The Grothendieck ring in geometry and topology, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 442–446. MR 0180975
- M. F. Atiyah and F. Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (1962), 25–45. MR 145560, DOI 10.1016/0040-9383(62)90094-0
- E. Ballico, F. Catanese, and C. Ciliberto (eds.), Classification of irregular varieties, Lecture Notes in Mathematics, vol. 1515, Springer-Verlag, Berlin, 1992. Minimal models and abelian varieties. MR 1180332, DOI 10.1007/BFb0098332
- Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 101–145. MR 412190
- Spencer Bloch, Algebraic cycles and values of $L$-functions. II, Duke Math. J. 52 (1985), no. 2, 379–397. MR 792179, DOI 10.1215/S0012-7094-85-05219-6
- S. Bloch and H. Esnault. The coniveau filtration and non-divisibility for algebraic cycles. Math. Ann. 304 (1996), 303-314.
- Jean-Louis Colliot-Thélène, Cycles algébriques de torsion et $K$-théorie algébrique, Arithmetic algebraic geometry (Trento, 1991) Lecture Notes in Math., vol. 1553, Springer, Berlin, 1993, pp. 1–49 (French). MR 1338859, DOI 10.1007/BFb0084728
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, Quelques gammes sur les formes quadratiques, J. Algebra 84 (1983), no. 2, 449–467 (French). MR 723402, DOI 10.1016/0021-8693(83)90088-1
- P. E. Conner and Larry Smith, On the complex bordism of finite complexes, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 117–221. MR 267571
- P. E. Conner and Larry Smith, On the complex bordism of finite complexes. II, J. Differential Geometry 6 (1971/72), 135–174. MR 298672
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
- Phillip A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460–495; 90 (1969), 496–541. MR 0260733, DOI 10.2307/1970746
- Masana Harada and Akira Kono, On the integral cohomology of extraspecial $2$-groups, Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 215–219. MR 885105, DOI 10.1016/0022-4049(87)90025-9
- Robin Hartshorne, Equivalence relations on algebraic cycles and subvarieties of small codimension, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1973, pp. 129–164. MR 0369359
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
- F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
- Kouichi Inoue, The Brown-Peterson cohomology of $B\textrm {SO}(6)$, J. Math. Kyoto Univ. 32 (1992), no. 4, 655–666. MR 1194107, DOI 10.1215/kjm/1250519400
- David Copeland Johnson and W. Stephen Wilson, Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327–353. MR 334257, DOI 10.1016/0040-9383(73)90027-X
- Akira Kono and Nobuaki Yagita, Brown-Peterson and ordinary cohomology theories of classifying spaces for compact Lie groups, Trans. Amer. Math. Soc. 339 (1993), no. 2, 781–798. MR 1139493, DOI 10.1090/S0002-9947-1993-1139493-4
- Peter S. Landweber, Coherence, flatness and cobordism of classifying spaces, Proc. Advanced Study Inst. on Algebraic Topology (Aarhus, 1970) Aarhus Univ., Matematisk Inst., Aarhus, 1970, pp. 256–269. MR 0271964
- Peter S. Landweber, Elements of infinite filtration in complex cobordism, Math. Scand. 30 (1972), 223–226. MR 326762, DOI 10.7146/math.scand.a-11077
- A. S. Merkur′ev and A. A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 1135–1136 (Russian). MR 675529
- John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. MR 99653, DOI 10.2307/1969932
- J. Milnor, On the cobordism ring $\Omega ^{\ast }$ and a complex analogue. I, Amer. J. Math. 82 (1960), 505–521. MR 119209, DOI 10.2307/2372970
- J. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337–341. MR 159327
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
- Madhav V. Nori, Algebraic cycles and Hodge-theoretic connectivity, Invent. Math. 111 (1993), no. 2, 349–373. MR 1198814, DOI 10.1007/BF01231292
- S. P. Novikov, Homotopy properties of Thom complexes, Mat. Sb. (N.S.) 57 (99) (1962), 407–442 (Russian). MR 0157381
- Kapil H. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math. (2) 139 (1994), no. 3, 641–660. MR 1283872, DOI 10.2307/2118574
- R. Parimala and V. Suresh, Zero-cycles on quadric fibrations: finiteness theorems and the cycle map, Invent. Math. 122 (1995), no. 1, 83–117. MR 1354955, DOI 10.1007/BF01231440
- Daniel Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293–1298. MR 253350, DOI 10.1090/S0002-9904-1969-12401-8
- Daniel Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971), 29–56 (1971). MR 290382, DOI 10.1016/0001-8708(71)90041-7
- Daniel Quillen, The $\textrm {mod}$ $2$ cohomology rings of extra-special $2$-groups and the spinor groups, Math. Ann. 194 (1971), 197–212. MR 290401, DOI 10.1007/BF01350050
- D. Ravenel, W. S. Wilson, and N. Yagita. Brown-Peterson cohomology from Morava $K$-theory. Preprint (1993).
- A. A. Rojtman, The torsion of the group of $0$-cycles modulo rational equivalence, Ann. of Math. (2) 111 (1980), no. 3, 553–569. MR 577137, DOI 10.2307/1971109
- Chad Schoen, On the computation of the cycle class map for nullhomologous cycles over the algebraic closure of a finite field, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 1, 1–50. MR 1305423
- Jean-Pierre Serre, Sur la topologie des variétés algébriques en caractéristique $p$, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 24–53 (French). MR 0098097
- Larry Smith, On realizing complex bordism modules. Applications to the stable homotopy of spheres, Amer. J. Math. 92 (1970), 793–856. MR 275429, DOI 10.2307/2373397
- Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858
- B. Totaro. Chow groups, Chow cohomology, and linear varieties. To appear in J. Alg. Geo.
- B. Totaro. The Chow ring of classifying spaces. Preprint.
- W. Stephen Wilson, The $\Omega$-spectrum for Brown-Peterson cohomology. II, Amer. J. Math. 97 (1975), 101–123. MR 383390, DOI 10.2307/2373662
Bibliographic Information
- Burt Totaro
- Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637
- MR Author ID: 272212
- Email: totaro@math.uchicago.edu
- Received by editor(s): May 28, 1996
- Received by editor(s) in revised form: December 2, 1996
- © Copyright 1997 American Mathematical Society
- Journal: J. Amer. Math. Soc. 10 (1997), 467-493
- MSC (1991): Primary 14C25; Secondary 55N22
- DOI: https://doi.org/10.1090/S0894-0347-97-00232-4
- MathSciNet review: 1423033