in [@Teh:real] which is defined by $RL _qH _n(X)=\pi _{n-q}(\mcal{R}_{q}(X))$ for $n\geq q$ and called reduced Lawson homology. In this notation our vanishing result reads $RL _qH _n(X)=0$ for any $n>\dim(X)$. Thus our vanishing result shows that the Friedlander-Mazur conjecture holds for the reduced Lawson homology of real varieties.

The homotopy groups of $\mcal{R}_{q}(X)$ fit into a long exact sequence $$\cdots \to \pi_{k+1}\mcal{R}_{q}(X) \to \pi_{k}\mcal{Z}_{q}(X_{\C})^{av} \to L_{q}H\R_{q-k,q}(X) \to \pi_{k}\mcal{R}_{q}(X) \to \cdots$$ where $L_{q}H\R_{q-k,q}(X) = \pi_{k}\mcal{Z}_{q}(X_{\C})^{G}$ is the real Lawson homology introduced by dos Santos in [@DS:real]. As a consequence of Suslin rigidity the homotopy groups of $\mcal{R}_{q}(X)$ are also related to motivic cohomology of $X$ $$\cdots  \to \pi_{k}z_{equi}(\A^{q}_{\C},0)(X_{\C}\times \Delta^{\bullet}_{\C})^{av} \to H_{\mcal{M}}^{2q-k}(X; \Z(q)) \to \pi_{k}\mcal{R}_{q}(X) \to \cdots .$$

Thus an immediate corollary of the vanishing theorem is an identification of the homotopy groups of the space of averaged cycles and of the complex of averaged equidimensional cycles.

Let $X$ be a smooth quasi-projective real variety. Then for any $k\geq \dim X -q+1$ $$L_{q}H\R_{q-k,q}(X) = \pi_{k}\mcal{Z}_{q}(X_{\C})^{av}$$ and $$H_{\mcal{M}}^{2q-k}(X;\Z(q)) = \pi_{k}z_{equi}(\A^{q}_{\C},0)(X_{\C}\times \Delta^{\bullet}_{\C})^{av}.$$

Theorem \[mth\] also implies that the mod-2 homotopy groups of the topological group of average cycles satisfy an optimal vanishing (see also Example \[avopt\]).

Let $X$ be a smooth projective real 