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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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A theory of algebraic cocycles
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by Eric M. Friedlander and H. Blaine Lawson PDF
Bull. Amer. Math. Soc. 26 (1992), 264-268 Request permission

Abstract:

We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a "cohomology theory" for complex algebraic varieties. The theory is bigraded, functorial, and admits Gysin maps. It carries a natural cup product and a pairing to L-homology. Chern classes of algebraic bundles are defined in the theory. There is a natural transformation to (singular) integral cohomology theory that preserves cup products. Computations in special cases are carried out. On a smooth variety it is proved that there are algebraic cocycles in each algebraic rational $(p,p)$-cohomology class.
References
  • Eric M. Friedlander, Algebraic cycles, Chow varieties, and Lawson homology, Compositio Math. 77 (1991), no. 1, 55–93. MR 1091892
    • E. Friedlander and H. B. Lawson, A theory of algebraic cocycles, Ann. of Math. (to appear).
  • Eric M. Friedlander and Barry Mazur, Filtrations on the homology of algebraic varieties, Mem. Amer. Math. Soc. 110 (1994), no. 529, x+110. With an appendix by Daniel Quillen. MR 1211371, DOI 10.1090/memo/0529
    • P. Lima-Filho, On a homology theory for algebraic varieties, IAS preprint, 1990. —, Completions and fibrations for topological monoids and excision for Lawson homology, Compositio Math., (1991).
  • H. Blaine Lawson Jr., Algebraic cycles and homotopy theory, Ann. of Math. (2) 129 (1989), no. 2, 253–291. MR 986794, DOI 10.2307/1971448
  • H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Algebraic cycles, Bott periodicity, and the Chern characteristic map, The mathematical heritage of Hermann Weyl (Durham, NC, 1987) Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 241–263. MR 974339, DOI 10.1090/pspum/048/974339
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 26 (1992), 264-268
  • MSC (2000): Primary 14C05; Secondary 14C30, 14F35
  • DOI: https://doi.org/10.1090/S0273-0979-1992-00269-0
  • MathSciNet review: 1118701