This work provides a detailed exposition of a classical topic from a very recent viewpoint. Friedlander and Mazur describe some foundational aspects of 'Lawson homology' for complex projective algebraic varieties, a homology theory defined in terms of homotopy groups of spaces of algebraic cycles. Attention is paid to methods of group completing abelian topological monoids. The authors study properties of Chow varieties, especially in connection with algebraic correspondences relating algebraic varieties. Operations on Lawson homology are introduced and analyzed. These operations lead to a filtration on the singular homology of algebraic varieties, which is identified in terms of correspondences and related to classical filtrations of Hodge and Grothendieck.
Introduction Questions and speculations Abelian monoid varieties Chow varieties and Lawson homology Correspondences and Lawson homology "Multiplication" of algebraic cycles Operations in Lawson homology Filtrations Appendix A. Mixed Hodge structures, homology, and cycle classes Appendix B. Trace maps and the Dold-Thom theorem Appendix Q. On the group completion of a simplicial monoid Bibliography.