  This 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.
