Let $X$ be a complex projective variety. The Chow monoid of $X$, denoted by $\mathcal{C}_p(X)$, is the monoid given by $p$-dimensional algebraic cycles of $X$. Considering a fixed embedding $i\colon X\subset\mathbb{P}^N$ we get a grading of $\mathcal{C}_p(X)$ given by the degree of cycles, i.e. $$\mathcal{C}_p(X)=\bigsqcup_{d\geq 0}\mathcal{C}_{p,d}(X)$$.

Notice that by considering on any $\mathcal{C}_{p,d}(X)$ the analytic topology we get a topology on the Chow monoid, that is the disjoint union topology.

The group pf $p$-cycles $\mathcal{Z}_p(X)$ can be obtained from $\mathcal{C}_p(X)$ as quotient $$\mathcal{Z}_p(X)=\mathcal{C}_p(X)\times\mathcal{C}_p(X)/\sim$$ where $(a,b)\sim(a',b')$ iff $a+b'=a'+b$. Therefore $\mathcal{C}_p(X)$ canonically induces a topology on $\mathcal{Z}_p(X)$, wich makes it a topological abelian group.

By a theorem due to J. Moore, a **connected** topological abelian group is homotopy equivalent to a product of Eilenberg-McLane spaces. Therefore, in order to justify the definition of Lawson homology groups $$\mathsf{L}_p\mathsf{H}_k(X):=\begin{cases} \pi_{k-2p}\mathcal{Z}_p(X), & \text{if} \: \: k\geq 2p\\
0, & \text{if} \: \: k< 2p\\
\end{cases},$$ one has to check that $\mathcal{Z}_p(X)$ is connected.

Any reference or hint for proving the connectedness of $\mathcal{Z}_p(X)$ is well accepted.

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