There are striking analogies between topology and arithmetic algebraic geometry, which studies the behavior of solutions to polynomial equations in arithmetic rings. One expression of these analogies is through the theory of etale cohomology, which is an algebraic version of topological cohomology. I will give an introduction to this circle of ideas and illustrate how they played the role in the recent resolution of a 1966 conjecture of Tate concerning a certain duality in the etale cohomology of 2-dimensional varieties over finite fields. This duality is analogous to the linking form in topology, and is analyzed via etale versions of Steenrod operations and characteristic classes. (The talk will not assume any familiarity with these concepts.)