  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.
