Algebraic cycles representing cohomology operations

Abstract

In this paper we will show that certain universal homology classes which are fundamental in topology are algebraic. To be specific, the products of Eilenberg–MacLane spaces \({\mathcal K}_{2q}\equiv K(\mathbf Z,{2}) \times K(\mathbf Z,{4}) \times \cdots \times K(\mathbf Z,{2q}) \) have models which are limits of complex projective varieties. Precisely, we have \({\mathcal K}_{2q}= \varinjlim \nolimits _{d\rightarrow \infty }\mathcal C^{q}_{d}(\mathbf P^{n})\) where \(\mathcal C^{q}_{d}(\mathbf P^{n})\) denotes the Chow variety of effective cycles of codimension q and degree d on \(\mathbf P_{\mathbf C}^{n}\). It is natural to ask which elements in the homology of \({\mathcal K}_{2q}\) are represented by algebraic cycles in these approximations. In this paper we find such representations for the even dimensional classes which are 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.