## Algebraic & Geometric Topology

### Simplicial complexes with lattice structures

George Bergman

#### 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 $Δ(L)$ (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

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

Dates
Revised: 6 May 2016
Accepted: 22 May 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841318

Digital Object Identifier
doi:10.2140/agt.2017.17.439

Mathematical Reviews number (MathSciNet)
MR3604382

Zentralblatt MATH identifier
1378.06004

#### Citation

Bergman, George. Simplicial complexes with lattice structures. Algebr. Geom. Topol. 17 (2017), no. 1, 439--486. doi:10.2140/agt.2017.17.439. https://projecteuclid.org/euclid.agt/1510841318

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