to be $$\mcal{Z}^{q}(W) = \frac{\Mor{\C}{W}{\mcal{C}_{0}(\P_{\C}^{q})}^{+}}{\Mor{\C}{W}{\mcal{C}_{0}(\P_{\C}^{q-1})}^{+}}$$

2.  Let $U$ be a quasi-projective real variety. The space of real algebraic $q$-cocyles is defined to be $$\mcal{Z}^{q}(U) = \frac{\Mor{\R}{U}{\mcal{C}_{0}(\P_{\R}^{q})}^{+}}{\Mor{\R}{U}{\mcal{C}_{0}(\P_{\R}^{q-1})}^{+}}$$

\[Gcocomm\] Let $U$ be a normal quasi-projective real variety then $$\frac{(\Mor{\C}{U_{\C}}{\mcal{C}_0(\P^q_{\C})}^{an,+})^{G}}{(
\Mor{\C}{U_{\C}}{\mcal{C}_0(\P^{q-1}_{\C})}^{an,+})^{G}} \xrightarrow{\iso} \left(\frac{\Mor{\C}{U_{\C}}{\mcal{C}_0(\P^q_{\C})}^{an,+}}{
\Mor{\C}{U_{\C}}{\mcal{C}_0(\P^{q-1}_{\C})}^{an,+}}\right)^{G}$$ and $$\frac{(\Mor{\C}{U_{\C}}{\mcal{C}_0(\P^q_{\C})}^{an,+})^{av}}{(
\Mor{\C}{U_{\C}}{\mcal{C}_0(\P^{q-1}_{\C})}^{an,+})^{av}} \xrightarrow{\iso} \left(\frac{\Mor{\C}{U_{\C}}{\mcal{C}_0(\P^q_{\C})}^{an,+}}{
\Mor{\C}{U_{\C}}{\mcal{C}_0(\P^{q-1}_{\C})}^{an,+}}\right)^{av}$$ are isomorphisms of topological groups.

By Lemma \[lemtop1\] and Proposition \[appcycagr\] it is enough to show that for $Y'\subseteq Y$ a closed subvariety of a projective real variety and $U$ a quasiprojective real variety that $$\frac{\mcal{C}_{r}(Y_{\C})(U_{\C})^{G}}{\mcal{C}_{r}(Y'_{\C})(U_{\C})^{G}}\xrightarrow{} \left(\frac{\mcal{C}_{r}(Y_{\C})(U_{\C})}{\mcal{C}_{r}(Y'_{\C})(U_{\C})}\right)^{G}$$ is an isomorphism of topological monoids.

We proceed in several steps.

1.  The map $$+: \mcal{C}_{r}(Y_{\C})(U_{\C})\times\mcal{C}_{r}(Y_{\C})(U_{\C}) \to \mcal{C}_{r}(Y_{\C})(U_{\C})$$ is a proper map. Observe that if $\alpha + \beta$ is equidimensional then both $\alpha$ and $\beta$ are equidimensional and therefore $$\xymatrix{
    \mcal{C}_{r}(Y_{\C})(U_{\C})\times\mcal{C}_{r}(Y_{\C})(U_{\C}) \ar[r]^-{+}\ar[d] & \mcal{C}_{r}(Y_{\C})(U_{\C}) \ar[d]\\
     \mcal{C}_{r+k}(U_{\C}\times Y_{\C})\times  \mcal{C}_{r+k}(U_{\C}\times Y_{\C}) \ar[r]^-{+} &  \mcal{C}_{r+k}(U_{\C}\times Y_{\C})
    }$$ is a pull-back square. Since addition is a proper map on effective cycles we see that it is a proper map for effective cocycles as well.

2.  The map $$\frac{\mcal{C}_{r}(Y_{\C})(U_{\C})^{G}}{\mcal{C}_{r}(Y'_{\C})(U_{\C})^{G}}\xrightarrow{} \left(\frac{\mcal{C}_{r}(Y_{\C})(U_{\C})}{\mcal{C}_{r}(Y'_{\C})(U_{\C})}\right)^{G}$$ is easily seen to be a continuous bijection by an argument similar to the one used in Lemma \[ccom\].

3.  Finally since $\mcal{C}_{r}(Y_{\C})(U_{\C}) \to \mcal{C}_{r}(Y_{\C})(U_{\C})/\mcal{C}_{r}(Y'_{\C})(U_{\C})$ is a closed map by Lemma 