Monodromy representations of completed coverings

Abstract

Abstract. In this paper we consider completed coverings that are branched coverings in the sense of Fox. For completed coverings between PL manifolds we give a characterization of the existence of a monodromy representation and the existence of a locally compact monodromy representation. These results stem from a characterization for the discreteness of a completed normal covering. We also show that completed coverings admitting a monodromy representations are discrete and that the image of the branch set is closed.

Local monodromy of branched covers and dimension of the branch set

“…Then f is a completed cover, that is, f is the unique extension of the covering For the general theory of these completions, see e.g. Fox [11], Berstein-Edmonds [2], Edmonds [8], or [1]. We call f ′ the regular part of f .…”

“…The monodromy triangle (1) of f : X → M is obtained as an extension of the monodromy triangle of its regular part f ′ : X ′ → M ′ . We refer to Berstein-Edmonds [2], or [1], for details.…”

“…Let f : M → N be a proper branched cover between n-manifolds. By a result of Berstein and Edmonds [2] (see also [1]), there exists a space X f and an action G f X f of the monodromy group G f of f by homeomorphisms for which the diagram…”

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.…”

“…Moreover (see e.g. [32]), nonconstant mappings of bounded distortion (1) are differentiable almost everywhere, (2) have positive Jacobian almost everywhere, and…”