American Journal of Mathematics

We show that a conjecture by Lawson holds, that is, the inclusion from the Chow variety $C_{p,d}({\Bbb P}^n)$ of all effective algebraic $p$-cycles of degree $d$ in $n$-dimensional projective space ${\Bbb P}^n$ to the space ${\mathcal C}_{p}({\Bbb P}^n)$ of effective algebraic $p$-cycles in ${\Bbb P}^n$ is $2d$-connected. As a result, the homotopy and homology groups of $C_{p,d}({\Bbb P}^n)$ are calculated up to $2d$. We also obtain the homotopy groups up to a certain level for the space of algebraic cycles with a fixed degree.