Strict orbifold atlases and weighted branched manifolds

Abstract

This note revisits the ideas in an earlier (2007) paper on orbifolds and branched manifolds, showing how the constructions can be simplified by using a version of the Kuranishi atlases recently developed by McDuff--Wehrheim. We first show that every orbifold has such an atlas, and then use it to obtain explicit models first for the nonsingular resolution of an oriented orbifold (which is a weighted nonsingular groupoid with the same fundamental class) and second for the Euler class of an oriented orbibundle. In this approach, instead of appearing as the zero set of a multivalued section, the Euler class is the zero set of a single-valued section of the pullback bundle over the resolution, and hence has the structure of a weighted branched manifold in which the weights and branching are canonically defined by the atlas.