particular the naive group completion of these groups are homotopy group completions.

Our main objects of interest are the group completions of submonoids of the Chow monoids of effective algebraic cycles on algebraic varieties. Let $k$ be a field of characteristic 0 and $j:Y\subseteq \P^n_k$ be a projective $k$-variety. The Chow variety $\mcal{C}_{q}(Y,j) = \coprod_{d\geq 0}\mcal{C}_{q,d}(Y,j) $ of effective $q$-dimensional cycles on $Y$ is an (infinite, disjoint) union of projective $k$-varieties. See [@F:ac] for details.

Cycle-spaces over $\C$
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Let $Z$ be a complex variety. Denote the set of complex points equipped with the analytic topology by $Z(\C)^{an}$. Since there will be little chance for confusion we will often simply write this space as $Z(\C)$ with the topology understood. If $j:Y\subseteq \P^n_{\C}$ is a projective variety then $\mcal{C}_q(Y,j)(\C)$ is a topological monoid and we will generally omit $j$ from the notation since the homeomorphism type of this space is independent of $j$. The monoid $\mcal{C}_q(Y)(\C)^{an}$ is tractable and therefore the naive group completion is a homotopy group completion. Write $$\mcal{Z}_q(Y) = (\mcal{C}_q(Y)(\C))^{+}$$ for the naive group group completion of this monoid. Define the filtration $\{0\} \subseteq\cdots \subseteq \mcal{Z}_{q,\leq d}(Y) \subseteq \mcal{Z}_{q,\leq d+1}(Y) \subseteq \cdots \subseteq \mcal{Z}_{q}(Y)$ by $$\mcal{Z}_{q,\leq d}(Y) = 