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

FAQ on the g-theorem and the hard Lefschetz theorem for face rings