  We reformulate the construction of Kontsevich's completion and use Lawson
homology to define many new motivic invariants. We show that the dimensions of
subspaces generated by algebraic cycles of the cohomology groups of two
$K$-equivalent varieties are the same, which implies that several conjectures
of algebraic cycles are $K$-statements. We define stringy functions which
enable us to ask stringy Grothendieck standard conjecture and stringy Hodge
conjecture. We prove a projective bundle theorem in morphic cohomology for
trivial bundles over any normal quasi-projective varieties.
