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 $\Delta \left(L\right)$ (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, $\Delta \left(M_3\right)$ 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 $\Delta \left(M_3\right)$ can be regarded as an example of this construction.