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

Karim Adiprasito

Published 2019-06-14Version 1

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 give a review over the two available proofs of the hard Lefschetz theorem for simplicial spheres, as well as recent work of Karu, Steinmeyer and the author, all of which share a common core introduced in arxiv:1812.10454: 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.