A Homological Lower Bound for Order Dimension of Lattices

We prove that if a nite lattice L has order dimension at most d, then the homology of the order complex of its proper part L vanishes in dimensions d ? 1 and higher. In case L can be embedded as a join-sublattice in N d then L actually has the homotopy type of a simplicial complex with d vertices. 1. Introduction. The order dimension OrderDim(P) of a nite partially ordered set (poset for short) P is deened to be the smallest positive integer d such that P is isomorphic to an induced subposet of a Cartesian product of d linear orders. OrderDim(P) turns out to be a very subtle and hard-to-compute invariant of P, with an extensive literature (see Tr]). Topological invariants of posets have also been studied extensively in the past few decades (see Bj] for some references). Here the basic object of study is the order complex of P, the abstract simplicial complex having the elements of P as its vertex set, and the linearly ordered subsets of P as its simplices. In what follows, we will abuse notation by making no distinction between the poset P, its order complex, and the topological space which is the geometric realization of this order complex. Given a nite simplicial complex X, deene its homological dimension HomDim(X) as follows: HomDim(X) := minfe : ~ H i (X; k) = 0 for all i > e; and for all elds kg where here ~ H i (X; k) refers to reduced simplicial homology of X with coeecients in k. In contrast to usual conventions, we set ~ H ?1 (X; k) = k for arbitrary X. We remark that simplicial homology is eeectively computable Mu, Chap. 1 x11], and hence so is HomDim(X). Our main result, Theorem 1, connects these two points of view in the case where the poset is a lattice, i.e. any two elements have a meet (greatest lower bound) and a join (least upper bound). There is some indication that the theory of order dimension may be better behaved for posets which are lattices than for arbitrary posets (see e.g. Tr, p. 69]). Our result gives a new lower bound for the order dimension of a nite lattice L, based on the topology of its proper part L , that is, the poset obtained by removing the bottom element ^ 0 and top element ^ 1 from …