Vanishing theorems for real algebraic cycles

We establish the analogue of the Friedlander-Mazur conjecture for Teh's reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety $X$ vanishes in homological degrees larger than the dimension of $X$ in all weights. As an application we obtain a vanishing of homotopy groups of the mod-$2$ topological groups of averaged cycles and a characterization in a range of indices of dos Santos' real Lawson homology as the homotopy groups of the topological group of averaged cycles. We also establish an equivariant Poincare duality between equivariant Friedlander-Walker real morphic cohomology and dos Santos' real Lawson homology. We use this together with an equivariant extension of the mod-$2$ Beilinson-Lichtenbaum conjecture to compute some real Lawson homology groups in terms of Bredon cohomology.