  In this paper we show that certain universal homology classes which are
fundamental in topology are algebraic. To be specific, the products of
Eilenberg-MacLane spaces ${\cal K}_{2q} \equiv K({\Bbb Z},2) \times K({\Bbb Z},
4) \times ... \times K({\Bbb Z}, 2q)$ have models which are limits of complex
projective varieties. Precisely, we have ${\cal K}_{2q} =
\lim_{d\to\infty}{\cal C}_d^q({\Bbb P}^n)$ where ${\cal C}_d^q({\Bbb P}^n)$
denotes the Chow variety of effective cycles of codimension $q$ and degree $d$
on ${\Bbb P}^n$. It is natural to ask which elements in the homology of ${\cal
K}_{2q}$ are represented by algebraic cycles in these approximations. In this
paper we find such representations for the even dimensional classes known as
Steenrod squares (as well as their Pontrjagin and join products). These classes
are dual to the cohomology classes which correspond to the basic cohomology
operations also known as the Steenrod squares.
