#### Volume 17, issue 1 (2017)

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Simplicial complexes with lattice structures

### George M Bergman

Algebraic & Geometric Topology 17 (2017) 439–486
##### 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.

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##### Keywords
order complex of a poset or lattice, topological lattice, distributive lattice, modular lattice, breadth of a lattice
##### Mathematical Subject Classification 2010
Primary: 06B30, 05E45
Secondary: 06A07, 57Q99