Simplicial complexes with lattice structures
arXiv:1602.00034 · doi:10.2140/agt.2017.17.439
Abstract
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). Lattice-theoretically, the resulting object is a subdirect product of copies of . We note properties of this construction and of some variants thereof, 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 obtained as a case of this construction.
Comments: 50 pages. Main change from version 2: Lemma 26 strengthened to include contractibility statement which in previous version was noted as "likely" in paragraph following that result. Several small typoes also corrected