and $k>0$. The cycle map $$\Phi:L^{q}H\R^{r,s}(X; \Z/2^{k}) \to H^{r,s}(X_{\C}(\C); \underline{\Z/2^{k}})$$ is an isomorphism if $r\leq 0$ (and $s\leq q$) and an injection if $r=1$ (and $s\leq q$).

Using Friedlander-Voevodsky duality for bivariant cycle theory we show in Corollary \[PD\] that the equivariant morphic cohomology and real Lawson homology groups are isomorphic through a Poincare duality. As a consequence the equivariant Beilinson-Lichtenbaum says that in a range we may compute the mod-2 real Lawson homology groups in terms of mod-2 Bredon cohomology. This allows a computation for curves with integral coefficients.

Let $X$ be a smooth real curve. Then $$L^{q}H\R^{r,s}(X; \Z) \to H^{r,s}(X_{\C}(\C); \underline{\Z})$$ is an isomorphism for any $q\geq 0$, $r\leq q$, and $s\leq q$.

The space of reduced cocycles on $X$, related via Poincare duality with the space of reduced cycles, is defined as $$\mcal{R}^{q}(X)=\frac{\mcal{Z}^{q}(X_{\C})^{G}}{\mcal{Z}^{q}(X_{\C})^{av}}$$ where $\mcal{Z}^{q}(X_{\C})$ is the space of algebraic cocycles on $X_{\C}$ and $\mcal{Z}^{q}(X_{\C})^{G}$ agrees with the space of real cocycles introduced by Friedlander-Walker in [@FW:real] (see Proposition \[nqp\]). There is a natural comparison map $$\label{cmap}
\mcal{R}^{q}(X) \to \map{X(\R)}{\mcal{R}_{0}(\A^{q})}$$ and since $\mcal{R}_{0}(\A^{q})=K(\Z/2,q)$ this provides a natural map $$\label{tcycm}
cyc _{k}: \pi_{k}\mcal{R}^{q}(X) \to H^{q-k}_{sing}(X(\R);\Z/2)$$ which is the cycle map for reduced morphic cohomology groups defined in [@Teh:real]. 