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). Lattice-theoretically, 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:
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arXiv e-prints
- Pub Date:
- January 2016
- DOI:
- 10.48550/arXiv.1602.00034
- arXiv:
- arXiv:1602.00034
- Bibcode:
- 2016arXiv160200034B
- Keywords:
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- Mathematics - Rings and Algebras;
- Mathematics - Algebraic Topology;
- 05E45;
- 06B30 (Primary);
- 06A07;
- 57Q99 (Secondary)
- E-Print:
- 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