cycles fixed by conjugation is the topological group $\mcal{Z}_{q}(X_{\C})^{av}$ of averaged cycles which are the cycles of the form $\alpha + \overline{\alpha}$. The space of reduced cycles on $X$ is the quotient topological group $$\mcal{R}_{q}(X) = \frac{\mcal{Z}_{q}(X_{\C})^{G}}{\mcal{Z}_{q}(X_{\C})^{av}}.$$ Homotopy groups of some of the above abelian topological groups are related to classical topological invariants. For example for $X$ a projective real variety we obtain the singular homology groups $\pi_{*}\mcal{R}_{0}(X)= H _*(X(\mathbb{R}),\mathbb{Z}/2)$ [@Teh:real] and $\pi_*\mcal{Z}_{0}(X_{\C})^{av}= H _*(X_{\C}(\mathbb{C})/G,\mathbb{Z})$ [@LLM:real], as well as Bredon homology $\pi_{*}\mcal{Z}_{0}(X) = H_{*,0}(X_{\C}(\C);\underline{\Z})$ [@LF:Gequiv]. Other homotopy groups are related to classical algebraic geometry invariants. For example $\pi _0(Z _r(X _\mathbb{C})^G)$ computes the group of algebraic cycles of dimension r on $X$ modulo real algebraic equivalence [@FW:real] and consequently with $\mathbb{Z}/n$ coefficients equals the Chow group $CH _r(X)\tensor\mathbb{Z}/n$ (see Proposition \[fcoeffagr\]). However most of these homotopy groups remain a mysterious combination of topological and algebraic information of the real variety $X$.

These homotopy groups are hard to compute and examples are few. Nonetheless an examination of existing computations shows that these homotopy groups are all zero in large degrees. For example, in ([@Lam:t]) Lam proves that $$\mcal{R}_{q}(\mathbb{P}^n _\mathbb{R})\simeq \prod _{i=0}^{n-q}K(\mathbb{Z}/2,i) .$$ In particular $\pi _k(\mcal{R}_{q}(\mathbb{P}^n _\mathbb{R}))=0$ for $k>n-q$. Similar vanishing 