Torsion algebraic cycles and étale cobordism

We prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of ℓ-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive characteristic.