  We show that, if the local dimension of the branch set of a discrete and open
mapping $f\colon M\to N$ between $n$-manifolds is less than $(n-2)$ at a point
$y$ of the image of the branch set $fB_f$, then the local monodromy of $f$ at
$y$ is perfect. In particular, for generalized branched covers between
$n$-manifolds the dimension of $fB_f$ is exactly $(n-2)$ at the points of
abelian local monodromy. As an application, we show that a generalized branched
covering $f\colon M \to N$ of local multiplicity at most three between
$n$-manifolds is either a covering or $fB_f$ has local dimension $(n-2)$.
