variety. Define the topological group of *reduced cocycles* to be the quotient topological group $$\mcal{R}^{q}(X) = \frac{\mcal{Z}^{q}(X_{\C})^{G}}{\mcal{Z}^{q}(X_{\C})^{av}} .$$ Lemma \[avl\] shows that $\mcal{R}^{q}(X)$ is a Hausdorff topological group.

In [@Teh:real] Teh defines $$\mcal{R}_{0}(Y)(X)=\frac{\Mor{\C}{X_{\C}}{\mcal{C}_{0}(Y_{\C})}^{+,G}}{ \Mor{\C}{X_{\C}}{\mcal{C}_{0}(Y_{\C})}^{+,av}}$$ and defines the reduced cocycles are defined as $$\mcal{R}^{q}(X) = \frac{\mcal{R}_{0}(\P^{q})(X)}{\mcal{R}_{0}(\P^{q-1})(X)}$$ for any real normal projective variety X and real projective variety Y.

By Proposition \[Gcocomm\] this definition and the one above give isomorphic topological groups.

\[avl\](c.f. [@Teh:real Proposition 2.4]) Let $X$ be a real projective variety. The subset of averaged cocycles $\mcal{Z}^{q}(X_{\C})^{av} \subseteq \mcal{Z}^{q}(X_{\C})$ is a closed subgroup.

Write $\overline{f}$ for $\sigma\cdot f$ and $\overline{V}$ for $\sigma\cdot V$. Suppose that $\{[f_n]+\overline{[f_{n}]}\}$ is a sequence in $\mcal{Z}^{q}(X_{\C})^{av}$ which converges in $\mcal{Z}^{q}(X_{\C})$. Write $[\gamma] = \lim_{n\to \infty} [f_{n}] + \overline{[f_{n}]}$ for its limit. We need to conclude that $[\gamma]$ is an averaged cocycle.

The set $\{[f_n] + [\overline{f_n}]\} \cup \{[\gamma]\}\subseteq \mcal{Z}^{q}(X_{\C})$ is compact. Applying the duality map to this set yields the compact subset $$\{\Gamma([f_n]) + \Gamma([\overline{f_n}])\} \cup \{\Gamma([\gamma])\} \subseteq 
\mcal{Z}_{d}( X_{\C}\times \A^{q}_{\C}).$$ Since this is a compact subset it lies in $\mcal{Z}_{d,\leq k}(X_{\C}\times 
\A^{q}_{\C})$ for some $k$.

The sequence $\{[g_{n}]\}\subseteq \mcal{Z}_{d,\leq k}(X_{\C}\times 
\A^{q}_{\C})$ has a convergent subsequence. Write $\{[g_{n_{i}}] \}$ for this convergent subsequence and write 