Non-manifold monodromy spaces of branched coverings between manifolds

TL;DR

This paper investigates the properties of monodromy spaces associated with branched coverings between manifolds, revealing that in three dimensions, these spaces can be non-manifold and lack local contractibility, unlike the two-dimensional case.

Contribution

The paper demonstrates that the monodromy space of a proper branched cover in three dimensions need not be a manifold, extending the understanding of branched covers beyond two dimensions.

Findings

Monodromy spaces can be non-manifold in 3D

Constructed a self-map of the 3-sphere with non-manifold monodromy space

Contrasts with the 2D case where monodromy spaces are manifolds

Abstract

By a construction of Berstein and Edmonds every proper branched cover f between manifolds is a factor of a branched covering orbit map from a locally connected and locally compact Hausdorff space called the monodromy space of f to the target manifold. For proper branched covers between 2-manifolds the monodromy space is known to be a manifold. We show that this does not generalize to dimension 3 by constructing a self-map of the 3-sphere for which the monodromy space is not a locally contractible space.