Local monodromy of branched covers and dimension of the branch set

Abstract

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

Some of the proofs have been streamlined and the title has been changed, but the content is the same as in the first version