  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.
