  New birational invariants for a projective manifold are defined by using
Lawson homology. These invariants can be highly nontrivial even for projective
threefolds. Our techniques involve the weak factorization theorem of Wlodarczyk
and tools developed by Friedlander, Lawson, Lima-Filho and others. A blowup
formula for Lawson homology is given in a separate section. As an application,
we show that for each n\geq 5, there is a smooth rational variety X of
dimension n such that the Griffiths groups Griff}_p(X) are infinitely generated
even modulo torsion for all p with 2\leq p\leq n-3.
