This is the third in a series of posts by Eran Nevo on the g-conjecture. Eran’s first post was devoted to the combinatorics of the g-conjecture and was followed by a further post by me on the origin of the g-conjecture. Eran’s second post was about the commutative-algebra content of the conjecture. It described the Cohen-Macaulay property (which is largely understood and known to hold for simplicial spheres) and the Lefshetz property which is known for simplicial polytopes and is wide open for simplicial spheres.
The g-conjecture and algebraic shifting
Back to the question from last time, Steinitz showed that
any simplicial 2-sphere is the boundary of a convex 3-polytope.
However, in higher dimension
there are many more simplicial spheres than simplicial polytopes,
on a fixed large number of vertices. Continue reading