The face numbers of homology spheres

The $g$-theorem is a momentous result in combinatorics that gives a complete numerical characterization of the face numbers of simplicial convex polytopes. The $g$-conjecture asserts that the same numerical conditions given in the $g$-theorem also characterizes the face numbers of all simplicial spheres, or even more generally, all simplicial homology spheres. In this paper, we prove the $g$-conjecture for simplicial $\mathbb{R}$-homology spheres. A key idea in our proof is a new algebra structure for polytopal complexes. Given a polytopal $d$-complex $\Delta$, we use ideas from rigidity theory to construct a graded Artinian $\mathbb{R}$-algebra $\Psi(\Delta,\nu)$ of stresses on a PL realization $\nu$ of $\Delta$ in $\mathbb{R}^d$, where overlapping realized $d$-faces are allowed. In particular, we prove that if $\Delta$ is a simplicial $\mathbb{R}$-homology sphere, then for generic PL realizations $\nu$, the stress algebra $\Psi(\Delta,\nu)$ is Gorenstein and has the weak Lefschetz property.