Simplicial complexes with lattice structures
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 $\Delta(L)$ (definition recalled). Latticetheoretically, the resulting object is a subdirect product of copies of $L$. We note properties of this construction and of some variants thereof, and pose several questions. For $M_3$ the $5$element nondistributive modular lattice, $\Delta(M_3)$ 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 $\Delta(M_3)$ can be obtained as a case of this construction.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1602.00034
 Bibcode:
 2016arXiv160200034B
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Algebraic Topology;
 05E45;
 06B30 (Primary);
 06A07;
 57Q99 (Secondary)
 EPrint:
 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