Kühn­el's Betti-number bounds for triangulated manifolds

Let $M$ be a combinatorial $d$-manifold with $n$ vertices and reduced Betti numbers $\tilde{\beta}_j(M)$ for $j=0,\dots,\lfloor d/2\rfloor$.

Kühn­el's conjecture. For $j=0,\dots,\lfloor(d-1)/2\rfloor$,

$$\binom{n-d+j-2}{j+1}\geq \binom{d+2}{j+1}\tilde{\beta}_j(M).$$

If $d$ is even, additionally

$$\binom{n-d/2-2}{d/2+1}\geq \binom{d+2}{d/2+1}\frac{\tilde{\beta}_{d/2}(M)}{2}.$$

If equality holds in one of these bounds for $j=s$, then $\tilde{\beta}_j(M)=0$ for $j\neq s$.

The conjecture extends known lower bounds for sphere products and the even-dimensional vertex bound. The general inequalities and their equality cases remain open.