Crosscut-simplicial Lattices

TL;DR

This paper introduces crosscut-simplicial lattices, characterizes their topological properties, and establishes their relation to SB-labellings and meet-semidistributivity, unifying several known lattice classes.

Contribution

It proves that all meet-semidistributive lattices are crosscut-simplicial and explores the equivalence of properties in chamber posets of hyperplane arrangements.

Findings

Meet-semidistributive lattices are crosscut-simplicial.

Interval complexes are either contractible or homotopy equivalent to spheres.

Equivalence of crosscut-simplicial and meet-semidistributive properties in chamber posets.

Abstract

We call a lattice crosscut-simplicial if the crosscut complex of every atomic 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 Meszaros introduced SB-labellings and proved that if a lattice has an SB-labelling then it is crosscut-simplicial. Some known examples of lattices with a natural SB-labelling 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-labelling. While not every crosscut-simplicial lattice is meet-semidistributive, we prove that these properties are equivalent for chamber posets of real hyperplane arrangements.