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Simplicial complexes with lattice structures

George M Bergman

Algebraic & Geometric Topology 17 (2017) 439–486

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.

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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
Received: 29 January 2016
Revised: 6 May 2016
Accepted: 22 May 2016
Published: 26 January 2017
George M Bergman
Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720-3840
United States