Local monodromy of branched covers and dimension of the branch set

TL;DR

This paper investigates the relationship between the local dimension of the branch set and the local monodromy in branched covers between manifolds, establishing conditions under which the monodromy is perfect and characterizing the branch set dimension.

Contribution

It proves that low-dimensional branch sets imply perfect local monodromy and characterizes the dimension of the branch set in generalized branched covers.

Findings

If the branch set dimension is less than n-2, the local monodromy is perfect.

For generalized branched covers, the branch set dimension is exactly n-2 at points with abelian monodromy.

A branched cover with multiplicity at most three is either a covering or has branch set dimension n-2.

Abstract

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 $(n-2)$.