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The $m$-Tamari lattices $\mathcal{T}_{n}^{(m)}$ were recently introduced byBergeron and Pr\'eville-Ratelle as posets on $m$-Dyck paths, and it was shownby Bousquet-M\'elou, Fusy and Pr\'eville-Ratelle that these lattices formintervals in the classical Tamari lattice $\mathcal{T}_{nm}$. It follows from atheorem 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 newEL-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 functionof $\mathcal{T}_{n}^{(m)}$, and we characterize the intervals of$\mathcal{T}_{n}^{(m)}$ according to their topological properties.

The Topology of the $m$-Tamari Lattices