Then $$\mcal{Z}_{r}(X_{\C})^{G} / \mcal{Z}_{r}(Y_{\C})^{G} \xrightarrow{\iso} (\mcal{Z}_{r}(X_{\C})/\mcal{Z}_{r}(Y_{\C}))^{G}$$ and $$\mcal{Z}_{r}(X_{\C})^{av} / \mcal{Z}_{r}(Y_{\C})^{av} \xrightarrow{\iso} (\mcal{Z}_{r}(X_{\C})/\mcal{Z}_{r}(Y_{\C}))^{av}$$ are isomorphisms of topological groups.

Consider the quotient maps $\pi: \mcal{Z}_{r}(X_{\C}) \to \mcal{Z}_{r}(X_{\C})/\mcal{Z}_{r}(Y_{\C})$ and $q:\mcal{Z}_{r}(X_{\C})^{G}\to \mcal{Z}_{r}(X_{\C})^{G}/\mcal{Z}_{r}(Y_{\C})^{G}$. Consider the filtration $\{(\pi\mcal{Z}_{r}(X_{\C})_{\leq d}\})^{G}$ of $(\mcal{Z}_{r}(X_{\C})/\mcal{Z}_{r}(Y_{\C}))^{G}$ and the filtration $\{q(\mcal{Z}_{r}(X_{\C})^{G}_{\leq d})\}$ of $\mcal{Z}_{r}(X_{\C})^{G} / \mcal{Z}_{r}(Y_{\C})^{G}$. These spaces have the weak topology given by these filtrations so it is enough to see that $$q (\mcal{Z}_{r}(X_{\C})_{\leq d}^{G})\xrightarrow{} (\pi\mcal{Z}_{r}(X_{\C})_{\leq d})^{G}$$ is a homeomorphism for all $d$.

First we show that $\mcal{Z}_{r}(X_{\C})_{\leq d}^{G}\to (\pi\mcal{Z}_{r}(X_{\C})_{\leq d})^{G}$ is surjective. If $[\eta]\in (\pi\mcal{Z}_{r}(X_{\C})_{\leq d})^{G}$, we can choose a representative $\eta= \sum n_{V} V\in \mcal{Z}_{r}(X_{\C})_{\leq d}$ such that each $V\nsubseteq Y_{\C}$. Since $\eta -\overline{\eta} \in \mcal{Z}_{r}(Y_{\C})$ (and each $\overline{V} \nsubseteq Y_{\C}$) we see that $\eta = \overline{\eta}$ and therefore the map $$\mcal{Z}_{r}(X_{\C})_{\leq d}^{G} \xrightarrow{} (\pi\mcal{Z}_{r}(X_{\C})_{\leq d})^{G}$$ is surjective. The map $$q\mcal{Z}_{r}(X_{\C})_{\leq d}^{G}\xrightarrow{} (\pi\mcal{Z}_{r}(X_{\C})_{\leq d})^{G}$$ is easily seen to be injective and since $\mcal{Z}_{r}(X_{\C})_{\leq d}^{G}$ is compact this map is closed so is a homeomorphism.

The second statement about the topological group of averaged cycles is proved in a similar fashion.

Spaces of algebraic cocycles
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In this section we recall the construction of topological monoids of algebraic cocycles [@FL:algco; @F:algco]. Let $X$, $Y$ be quasi-projective real varieties over $k=\C$ 