closure with closed complement $W_{\infty} = \overline{W}\backslash W$. This topology may also be described as follows. Let $$\mcal{E}_{r}(Z)(W) \subseteq \mcal{C}_{r+d}(\overline{W}\times Z)$$ denote the constructable submonoid consisting of effective cycles whose restriction to $W\times Z$ is equidimensional of relative dimension $r$ over $W$. By [@F:algco Proposition 1.8] the topology on $\mcal{C}_{r}(Z)(W)$ (given by the subspace topology above) coincides with the quotient topology given by $$\mcal{C}_{r}(Z)(W) = \frac{\mcal{E}_{r}(Z)(W)}{\mcal{C}_{r+d}(W_{\infty}\times Z)}.$$

Define the topological group of equidimensional cycles of relative dimension r over $W$ as $\mcal{Z}_{r}(Z)(W) = [\mcal{C}_{r}(Z)(W)]^{+}$ (where as usual the naive group completion is given the quotient topology). Since $\mcal{C}_{r}(Z)(W)$ is a tractable monoid the naive group completion is a homotopy group completion.

In [@F:ac] it is shown that a morphism of varieties $f:W\to \mcal{C}_{r}(Z)$ has an associated graph in $\mcal{Z}_{f} \in \mcal{C}_{r}(Z)(W)$. By [@FL:dual Proposition A.1] this defines an isomorphism of topological monoids $ \Gamma: \Mor{\C}{W}{\mcal{C}_r(Z)} \to \mcal{C}_{r}(Z)(W)$ for any normal, quasi-projective complex variety $W$ by [@FL:dual Proposition A.1]. Therefore the graph map $\Gamma$ also induces an isomorphism of topological abelian groups $$\Gamma: \Mor{\C}{W}{\mcal{C}_0(Z)}^{+} \to \mcal{Z}_{0}(W)(Z),$$ for any normal, quasi-projective variety $W$ and any projective variety $Z$. The composite of $\Gamma$ and the continuous inclusion $\mcal{Z}_{0}(W)(Z)\subseteq \mcal{Z}_{\dim W}(W\times Z)$ defines the 