Algebraic & Geometric Topology

Simplicial complexes with lattice structures

George Bergman

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If L is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex Δ(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 M3 the 5–element nondistributive modular lattice, Δ(M3) 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 Δ(M3) can be regarded as an example of this construction.

Article information

Algebr. Geom. Topol., Volume 17, Number 1 (2017), 439-486.

Received: 29 January 2016
Revised: 6 May 2016
Accepted: 22 May 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 06B30: Topological lattices, order topologies [See also 06F30, 22A26, 54F05, 54H12] 05E45: Combinatorial aspects of simplicial complexes
Secondary: 06A07: Combinatorics of partially ordered sets 57Q99: None of the above, but in this section

order complex of a poset or lattice topological lattice distributive lattice modular lattice breadth of a lattice


Bergman, George. Simplicial complexes with lattice structures. Algebr. Geom. Topol. 17 (2017), no. 1, 439--486. doi:10.2140/agt.2017.17.439.

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