I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of algebraic geometry.
In this community wiki question I would like to gather examples, in geometrical fields such as algebraic topology and algebraic geometry, of phenomena that manifest themselves by the presence of torsion in (co)homology groups and whose trace is consistently lost if we simply disregard the torsion part of those groups.
As guidelines for the answers:
Which kind of information is lost disregarding torsion in (co)homology? (provide examples)
What does the torsion part of (co)homology tell us about the geometric object involved?
(provide examples)
Here "(co)homology" should be understood in any relevant sense, from singular cohomology of cw complexes to étale cohomology of algebraic varieties and so on and so forth.
It may well be true that the algebro geometric examples have nothing to do, conceptually, with the topological ones: I'm not interested in a unifying pattern per se, but if such a unifying pattern does appear in some answers, well, it's just good.
Best Answer
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.