The g-conjecture for simplicial spheres

Let $L$ be a simplicial sphere, with $g$-vector defined from its $h$-vector by $g_0=h_0$ and $g_i=h_i-h_{i-1}$ for $i\geq 1$. An $M$-sequence is the Hilbert function of a standard graded Artinian algebra. The $g$-conjecture. The $g$-vector of $L$ is an $M$-sequence. This is the part of the $g$-conjecture that remains open; the paper proves the corresponding assertion for barycentric subdivisions of homology spheres and more general shellable complexes.