On the Topology of the Cambrian Semilattices

For an arbitrary Coxeter group $W$, David Speyer and Nathan Reading defined Cambrian semilattices $C_{\gamma }$ as semilattice quotients of the weak order on $W$ induced by certain semilattice homomorphisms. In this article, we define an edge-labeling using the realization of Cambrian semilattices in terms of $\gamma$-sortable elements, and show that this is an EL-labeling for every closed interval of $C_{\gamma }$. 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 Nathan Reading.