Local monodromy of branched covers and dimension of the branch set

Abstract

Abstract. We show that, if the local dimension of the image of the branch set of a discrete and open mapping f : M → 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 → N of local multiplicity at most three between n-manifolds is either a covering or f B f has local dimension (n − 2).

Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces

“…Then spt ω ∩ f −1 (p) = ∅ and therefore all the terms in the sum defining f # ω(p) are zero. This proves (1).…”

“…When f is a BLD-map, the group G acts on X f by bilipschitz maps, and f and ϕ are BLD-maps. See [1] and the references therein for details on monodromy representations.…”

Open and discrete maps with piecewise linear branch set images are piecewise linear maps

“…It is currently not known if such examples exist in lower dimensions. For example, the Church-Hemmingsen conjecture asks if there exists an open and discrete map in three dimensions with a branch set homeomorphic to a Cantor set (see [6] and [1]). In general the structure of the branch set of an open and discrete map, or even a quasiregular mapping, is not well understood but the topic garners great interest.…”

“…This example shares many of the properties of open and discrete maps with contained in an ‐simplicial complex, but it is not a PL mapping. For further discussion on this map see, for example, [1].…”

Mappings of finite distortion: Size of the branch set

“…These sets have very rich and complex topological and geometrical structures both of which have been studied in a variety of different contexts, see e.g. [1,2,12,13,14,25,35] and references there.…”