On the Topology of the Cambrian Semilattices

For an arbitrary Coxeter group W , Reading and Speyer defined Cambrian semilattices Cγ as sub-semilattices of the weak order onW induced by so-called γ-sortable elements. In this article, 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.