Connectivity of h-complexes

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.