  We construct rational projective 4-dimensional varieties with the property
that certain Lawson homology groups tensored with Q are infinite dimensional
Q-vector spaces. More generally, each pair of integers p and k, with k\geq 0,
p>0, we find a projective variety Y, such that L_pH_{2p+k}(Y) is infinitely
generated. We also construct two singular rational projective 3-dimensional
varieties Y and Y' with the same homeomorphism type but different Lawson
homology groups, specifically L_1H_3(Y) is not isomorphic to L_1H_3(Y')even up
to torsion.
