Algebraic & Geometric Topology

Simplicial complexes with lattice structures

George Bergman

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

Permanent link to this document
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

Subjects
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

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

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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