Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 17, Number 1 (2017), 439-486.
Simplicial complexes with lattice structures
If is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of . We note properties of this construction and of some variants, and pose several questions. For the –element nondistributive modular lattice, 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 can be regarded as an example of this construction.
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
Permanent link to this document
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
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