On the topology of the cambrian semilattices

Abstract

For an arbitrary Coxeter groupW and a Coxeter element γ ∈ W, Reading and Speyer defined the Cambrian semilattice Cγ as the sub-semilattice of the weak order on W induced by so-called sub-sortable elements. In this note, we define an edge-labeling of Cγ, and show that this is an EL-labeling for every closed interval of Cγ. In addition, we use our labeling to show that every finite open interval in a Cambrian semilattice is either contractible or spherical, and we characterize the spherical intervals, generalizing a result by Reading.