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We generalize the Harnack-Thom theorem to relate the ranks of the Lawsonhomology groups with $\Z_2$-coefficients of a real quasiprojective variety withthe ranks of its reduced real Lawson homology groups. In the case of zero-cyclegroup, we recover the classical Harnack-Thom theorem and generalize theclassical version to include real quasiprojective varieties. We use Weil'sconstruction of Picard varieties to construct reduced real Picard groups, andMilnor's construction of universal bundles to construct some weak models ofclassifying spaces of some cycle groups. These weak models are used to producelong exact sequences of homotopy groups which are the main tool in computingthe homotopy groups of some cycle groups of divisors. We obtain somecongruences involving the Picard number of a nonsingular real projectivevariety and the rank of its reduced real Lawson homology groups of divisors.

Harnack-Thom theorem for higher cycle groups and Picard varieties