Order

, Volume 34, Issue 3, pp 465–477 | Cite as

Crosscut-Simplicial Lattices

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Abstract

We call a finite lattice crosscut-simplicial if the crosscut complex of every nuclear interval is equal to the boundary of a simplex. Every interval of such a lattice is either contractible or homotopy equivalent to a sphere. Recently, Hersh and Mészáros introduced SB-labelings and proved that if a lattice has an SB-labeling then it is crosscut-simplicial. Some known examples of lattices with a natural SB-labeling include the join-distributive lattices, the weak order of a Coxeter group, and the Tamari lattice. Generalizing these three examples, we prove that every meet-semidistributive lattice is crosscut-simplicial, though we do not know whether all such lattices admit an SB-labeling. While not every crosscut-simplicial lattice is meet-semidistributive, we prove that these properties are equivalent for chamber posets of real hyperplane arrangements.

Keywords

Crosscut complex SB-labeling Semidistributive lattice Hyperplane arrangement 
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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA

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