*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*

### Squeezed spheres

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