Simplicial complexes with lattice structures

Abstract

If L is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex ∆(L) (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of L. We note properties of this construction and of some variants, and pose several questions. For M 3 the 5-element nondistributive modular lattice, ∆(M 3 ) is modular, but its underlying topological space does not admit a structure of distributive lattice, answering a question of Walter Taylor.We also describe a construction of "stitching together" a family of lattices along a common chain, and note how ∆(M 3 ) can be regarded as an example of this construction.05E45, 06B30; 06A07, 57Q99 A lattice structure on ∆(L)I came upon the construction studied here from a direction unrelated to the concept of order complex; so I will first motivate it in roughly the way I discovered it, then recall the order complex construction, which turns out to describe the topological structures of these lattices.

Taxotopy Theory of Posets I: van Kampen Theorems

“…One can think of a monotone map in Pos(P ⊔ P, P ) as a pair of maps in Pos(P, P ). But the taxotopy order in Λ(P ⊔ P, P ) is not the full product order in Λ(P ) 2 . In fact it is isomorphic to Fold(Λ(P ) 2 , ⪯ o ) where ⪯ o is the order obtained from the open book condition.…”

“…The complexes K and ∆(P(K)) are homotopy equivalent but unfortunately the partial order structures on P and P(∆(P )) are unrelated. Bergman [2] recently provided a way to lift the partial order on P to a partial order structure on the simplicial complex ∆(P ). Building on this idea, we will provide some results on simplicial taxotopy theory in the next paper.…”