2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69218This paper verifies a conjecture of Edelman and Reiner regarding the homology of the $h$-complex of a Boolean algebra. A discrete Morse function with no low-dimensional critical cells is constructed, implying a lower bound on connectivity. This together with an Alexander duality result of Edelman and Reiner implies homology-vanishing also in high dimensions. Finally, possible generalizations to certain classes of supersolvable lattices are suggested.Combinatorics05A05; 05E25text