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In their paper "Some Elementary Examples of Unirational Varieties Which are Not Rational", Artin and Mumford show that the torsion in $H^3(V, Z)$ of a non singular projective 3-fold $V$ is a birational invariant. This is great because it gives a cohomological obstruction to rationality (there is no torsion in the cohomology of projective space). They they are able to show that certain conic bundles over rational surfaces are not rational by exhibiting such torsion (their conic bundles are unirational, hence the title). The paper is very nice.