Generalised lower bound conjecture for triangulated manifolds

Let $M$ be a connected triangulated closed $d$-manifold, with $g$-numbers $g_i(M)$ and Betti numbers $\beta_i(M)$. For $1\leq l\leq \frac{d-1}{2}$, consider the lower-bound inequality

$$g_{l+1}(M)\geq \binom{d+2}{l+1}\sum_{i=1}^{l}(-1)^{l-i}\beta_i(M).$$

Generalised lower bound conjecture for triangulated manifolds. The displayed inequality holds, and equality for some $l<\frac{d-1}{2}$ holds if and only if $M\in {\cal K}_l(d)$. This includes the generalised lower bound conjecture for homology spheres; the $l=1$ case was known as a conjecture of Kalai and was proved under an orientability hypothesis, while the general case remains open.