\[qmcl\] we conclude that the continuous bijection $$\frac{\mcal{C}_{r}(Y_{\C})(U_{\C})^{G}}{\mcal{C}_{r}(Y'_{\C})(U_{\C})^{G}}\xrightarrow{} \left(\frac{\mcal{C}_{r}(Y_{\C})(U_{\C})}{\mcal{C}_{r}(Y'_{\C})(U_{\C})}\right)^{G}$$ is a closed map and therefore a topological isomorphism.

The second statement for average cocycles is proved in a similar fashion, using Proposition \[avl\] and that $C _r(Y _{\C})(U _{\C})^{av}\subseteq C _r(Y_{\C})(U _{\C})$ is closed.

As with the topological group of cycles on a real variety $X$ we may view the topological group of real cocycles as the topological group of cycles on the complexification which are fixed by the Galois action.

\[nqp\] Let $X$ be a normal quasi-projective real variety. Then $$\mcal{Z}^{q}(X) = \mcal{Z}^{q}(X_{\C})^{G}.$$

This follows from Proposition \[appcycagr\] together with the previous proposition since $\Mor{\R}{X}{\mcal{C}_0(\P^q_{\R})} = \Mor{\C}{X_{\C}}{\mcal{C}_0(\P^q_{\C})}^{G}$.

The space $\mcal{Z}^{q}(X_{\C})$ has the equivariant homotopy type of a $G$-$CW$-complex (see Corollary \[ehtype\]).

When $W=X_{\C}$ is the complexification of a quasi-projective real variety and $Z = Y_{\C}$ is the complexification of a projective real variety the graph map is an equivariant morphism. In particular the duality map $$\mcal{D}:\mcal{Z}^{q}(X_{\C})\to \mcal{Z}_{d}(X_{\C}\times \A^{q}_{\C})$$ is an equivariant continuous map.

1.  Let $X$ be a projective real variety. Define the topological group of *averaged cocycles* to be $$\mcal{Z}^{q}(X_{\C})^{av} = \{f + \sigma \cdot f | f\in \mcal{Z}^{q}(X_{\C})\}\subseteq \mcal{Z}^{q}(X_{\C}).$$

2.  Let $X$ be a normal projective real 