Let $X\subset\mathbb{P}^N:=\mathbb{P}^N(\mathbb{C})$ be a projective variety and denote by $X(p)$ the set of $p$-dimensional subvariety of $X$. The free abelian group generated by $X(p)$ is the space of $p$-cycles on $X$. We will denote it by $\mathcal{Z}_p(X)$. The monoid (w.r.t. the formal sum $+$) generated by $X(p)$ is called Chow monoid and we will denote it by $\mathcal{C}_p(X)$.

It's very well known that it's possible to put a topology on $\mathcal{C}_p(X)$. Therefore, since $$\mathcal{Z}_p(X)=\mathcal{C}_p(X)\times\mathcal{C}_p(X)/\sim,$$ where $(x,y)\sim (x',y')$ if $x+y'=x'+y$, we btain a quotient topology on the group of $p$-cycles. This last fact allow us to define the Lawson homology groups of $X$ as $$\mathsf{L}_m\mathsf{H}_\ell(X):=\begin{cases}\pi_{\ell-2m} \, \mathcal{Z}_m(X), \textit{ if } \ell\geq 2m\\ 0, \textit{ otherwise }\end{cases}. $$

See for example this or the paper Lawson, B.: Spaces of algebraic cycles, in Surveys in Differential Geometry 2 for more details.

I would know if this is really an homology theory, in the sense of Eilenberg and Steenrod. Any reference is appreciated.

NB: I apologize in advance if my question could be almost trivial but I'm not a professional mathematician, I'm studying the Lawson homology for my master degree thesis.


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