# Motivation of Lawson Homology

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.

• Did you intend to add the hypothesis that the degree of $a$ equals the degree of $b$? If not, then $\text{degree}(a)-\text{degree}(b)$ defines a locally constant function on $\mathcal{Z}_p(X)$ that is not constant, hence $\mathcal{Z}_p(X)$ is not connected. May 8, 2018 at 18:22
• I'm using the definition of group completion given by Lawson at pag. 148 of the paper Spaces of Algebraic Cycles. In the same paper the author says (pag. 157, beginnig of par.8) that, by the theorem cited above, the homotopy type of $\mathcal{Z}_p(X)$ is that of a product of E-M spaces. Paper: intlpress.com/site/pub/files/_fulltext/journals/sdg/1993/0002/… May 8, 2018 at 18:36
• Did you read Corollary 8.2 of that article? Lawson does not claim that $\mathcal{Z}_p(X)$ is connected. Quite the contrary, he explicity describes the group of connected components. May 8, 2018 at 19:06
• I might be wrong, but if G is a (locally path connected) topological group and G_0 is the connected component of the identity, isn't there a (noncanonical) homeomorphism $G\cong G_0 \times \pi_0G$ given by choosing a point in each component, hence $G$ is a product of E-ML spaces? More homotopically, $\mathrm[{Sing}G$ is a simplicial abelian group, hence it's the homotopy type of a chain complex and so a product of E-ML spaces. May 8, 2018 at 19:32
• @JasonStarr You are right. What I dont' understand is why the homotopy type of $\mathcal{Z}_p(X)$ is completely determined by the homotopy groups $\pi_\ast\mathcal{Z}_p(X)$ by using the theorem 3.4 (in the paper). May 8, 2018 at 20:19