A note on Lefschetz spheres and their relatives

Inspired by the works of Adiprasito, Babson, Nevo, and Murai on the $g$-conjecture, we consider different classes of PL-spheres and the relations between them. We focus on a certain class of spheres that is in the intersection of vertex-decomposable spheres (a concept due to Provan and Billera) and strongly edge-decomposable spheres (a concept due to Eran Nevo). The spheres in this class are exactly those vertex-decomposable ones for which Adiprasito's recent proof method works.