Johan Strååt: Higher order complexes of distributive lattices

The k-th order complex of a poset is the family of unions of k chains. (Equivalently, a simplex is a set of elements that does not contain an antichain of cardinality k+1.) Björner showed many years ago that the k-th order complex of a planar distributive lattice is always shellable. I will show that this does not generalize to distributive lattices of higher dimension, not even for the second-order complex of a three-dimensional distributive lattice. Furthermore, it turns out that the second-order complex of a (truncated) Boolean lattice of even rank at least 4 is a non-orientable pseudomanifold.