  This short review is the result of a minicourse at the Sapienza University of
Rome the author gave about the proof of the $g$-theorem. We review the hard
Lefschetz theorem for simplicial spheres, as well as the theory at its core:
perturbations of maps, biased Poincar\'e pairings and a cobordism argument that
relates the Lefschetz property of a manifold to the Lefschetz property on its
boundary. We also sketch an alternative argument based on edge-contractions.
