On the Homological Dimension of Lattices

Let L be a finite lattice and let L ̂ = L − {0̂, 1̂}. It is shown that if the order complex ∆(L̂) satisfies H̃k−2 ∆(L̂)) 6 = 0 then |L | ≥ 2k. Equality |L | = 2k holds iff L is isomorphic to the Boolean lattice {0, 1}k.