Edelman–Reiner conjecture on the homology of Boolean algebra h-complexes

Following the standard shelling of the order complex of the truncated Boolean algebra $B_n-\{\hat{0},\hat{1}\}$, let $\Delta_n$ be the resulting $h$-complex. Edelman–Reiner conjecture. The reduced homology group $\widetilde{H}_i(\Delta_n,\mathbb{Z})$ is nonzero if and only if

$$\frac{3i+5}{2}\leq n\leq 3i+4.$$

This conjecture concerns the precise dimensions in which the $h$-complex of a Boolean algebra has nonvanishing reduced homology. The paper verifies the conjecture by constructing a discrete Morse function with no low-dimensional critical cells and combining the resulting connectivity bound with an Alexander duality result of Edelman and Reiner, which also gives homology vanishing in high dimensions.