Algebraic Cycles and Lawson Homology: an Application of Homotopy Theoretic Methods in Algebraic Cycles Theory

The fundamental object of interest in algebraic geometry is the structure of spaces of algebraic cycles on a projective variety. Any profound understanding of this will be very helpful to know the structure of projective manifolds. The study of algebraic cycles may date back to 1930s. A breakthrough of the homotopy theoretic approach to algebraic cycles is the Algebraic Suspension Theorem proved by Blaine Lawson in the late 1980s. This method has been developed by Eric Friedlander, Blaine Lawson and others. This book studies further properties of Lawson homology as well as relations to the singular homology and Chow groups. In particular, new nontrivial birational invariants for complex smooth projective varieties are defined using Lawson homology; Birational invariant statements for 1-cycles and codimension two cycles are given; Generalized Abel-Jacobi map for Lawson homology is constructed; Examples of both smooth and singular projective varieties are constructed to hold infinitely generated Lawson homology groups even up to torsion. It is suitable for those interested in complex algebraic geometry, especially the homotopy theoretic aspect of algebraic cycles theory.