  In this paper we introduce the Fourier-Mukai transform for Lawson homology of
abelian varieties and prove an inversion theorem for the Lawson homology as
well as the morphic cohomology of abelian varieties. As applications, we obtain
the direct sum decomposition of the Lawson homology and the morphic cohomology
groups with rational coefficients, inspired by Beauville's works on the Chow
theory. An analogue of the Beauville conjecture for Chow groups is proposed and
is shown to be equivalent to the (weak) Suslin conjecture for Lawson homology.
A filtration on Lawson homology is proposed and conjecturally it coincides to
the filtration given by the direct sum decomposition of Lawson homology for
abelian varieties. Moreover, a refined Friedlander-Lawson duality theorem is
obtained for abelian varieties. We summarize several related conjectures in
Lawson homology theory in the appendix for convenience.
