The Topology of the $m$-Tamari Lattices

The $m$-Tamari lattices ${\mathcal{T}}_n^{(m)}$ were recently introduced by Bergeron and Pr'eville-Ratelle as posets on $m$-Dyck paths, and it was shown by Bousquet-M'elou, Fusy and Pr'eville-Ratelle that these lattices form intervals in the classical Tamari lattice ${\mathcal{T}}_{nm}$. It follows from a theorem by Bj"orner and Wachs and a basic property of EL-shellable posets, that the $m$-Tamari lattices are EL-shellable. In this article, we define a new EL-labeling of the $m$-Tamari lattices completely in terms of $m$-Dyck paths. With the help of this labeling, we compute the values of the M"obius function of ${\mathcal{T}}_n^{(m)}$, and we characterize the intervals of ${\mathcal{T}}_n^{(m)}$ according to their topological properties.