A theory of algebraic cocycles

Authors:
Eric M. Friedlander and H. Blaine Lawson

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

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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 -cohomology class.

**[F]**E. Friedlander,*Algebraic cycles, Chow varieties, and Lawson homology*, Compositio Math.**77**(1991), 55-93. MR**1091892 (92a:14005)****[FL]**E. Friedlander and H. B. Lawson,*A theory of algebraic cocycles*, Ann. of Math. (to appear).**[FM]**E. Friedlander and B. Mazur,*Filtrations on the homology of algebraic varieties*(to appear). MR**1211371 (95a:14023)****[L-F1]**P. Lima-Filho,*On a homology theory for algebraic varieties*, IAS preprint, 1990.**[L-F2]**-,*Completions and fibrations for topological monoids and excision for Lawson homology*, Compositio Math., (1991).**[L]**H. B. Lawson, Jr.,*Algebraic cycles and homotopy theory*, Ann. of Math. (2)**129**(1989), 253-291. MR**986794 (90h:14008)****[LM]**H. B. Lawson, Jr. and M.-L. Michelsohn,*Algebraic cycles, Bott periodicity, and the Chern characteristic map*, The Mathematical Heritage of Herman Weyl, Amer. Math. Soc., Providence, RI, 1988, pp. 241-264. MR**974339 (90d:14010)**

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Additional Information

DOI:
https://doi.org/10.1090/S0273-0979-1992-00269-0

Keywords:
Algebraic cycle,
Chow variety,
algebraic cocycle,
cohomology

Article copyright:
© Copyright 1992
American Mathematical Society