On the Topology of the Cambrian Semilattices

TL;DR

This paper introduces an edge-labeling for Cambrian semilattices associated with Coxeter groups, proving topological properties of their intervals and generalizing previous results on their structure.

Contribution

It defines an EL-labeling for Cambrian semilattices and characterizes the topological nature of their intervals, extending prior work by Nathan Reading.

Findings

Established an EL-labeling for Cambrian semilattices.

Proved that finite open intervals are either contractible or spherical.

Characterized the spherical intervals within Cambrian semilattices.

Abstract

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.