Algebraic Cycles Representing Cohomology Operations

TL;DR

This paper demonstrates that certain universal homology classes in topology, specifically Steenrod squares, can be represented by algebraic cycles within limits of complex projective varieties, linking algebraic geometry and topology.

Contribution

It shows that fundamental topological classes like Steenrod squares are algebraic and can be realized within algebraic cycles in projective varieties.

Findings

Steenrod squares are represented by algebraic cycles in Chow varieties.

Homology classes in products of Eilenberg-MacLane spaces are algebraic.

Limits of Chow varieties model these topological spaces.

Abstract

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.