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

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]**Eric M. Friedlander,*Algebraic cycles, Chow varieties, and Lawson homology*, Compositio Math.**77**(1991), no. 1, 55–93. MR**1091892****[FL]**E. Friedlander and H. B. Lawson,*A theory of algebraic cocycles*, Ann. of Math. (to appear).**[FM]**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**, 10.1090/memo/0529**[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. Blaine Lawson Jr.,*Algebraic cycles and homotopy theory*, Ann. of Math. (2)**129**(1989), no. 2, 253–291. MR**986794**, 10.2307/1971448**[LM]**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**, 10.1090/pspum/048/974339

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Additional Information

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

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

Article copyright:
© Copyright 1992
American Mathematical Society