  We study in detail the so-called Chow-weight homology of Voevodsky motivic
complexes and relate it to motivic homology. We generalize earlier results and
prove that the vanishing of higher motivic homology groups of a motif $M$
implies similar vanishing for its Chow-weight homology along with effectivity
properties of the higher terms of its weight complex $t(M)$ and of higher
Deligne weight quotients of its cohomology. Applying this statement to motives
with compact support we obtain a similar relation between the vanishing of Chow
groups and the cohomology with compact support of varieties. Moreover, we prove
that if higher motivic homology groups of a geometric motif or a variety over a
universal domain are torsion (in a certain "range") then the exponents of these
groups are uniformly bounded.
  To prove our main results we study Voevodsky slices of motives. Since the
slice functors do not respect the compactness of motives, the results of the
previous Chow-weight homology paper are not sufficient for our purposes; this
is our main reason to extend them to ($w_{Chow}$-bounded below) motivic
complexes.
