Strict orbifold atlases and weighted branched manifolds

TL;DR

This paper simplifies the theory of orbifolds and branched manifolds using Kuranishi atlases, providing explicit models for resolutions and Euler classes with canonical weights and branching.

Contribution

It demonstrates that every orbifold admits a Kuranishi atlas and constructs explicit models for resolutions and Euler classes as weighted branched manifolds.

Findings

Every orbifold has a Kuranishi atlas.

Explicit models for nonsingular resolutions of orbifolds.

Euler classes can be represented as zero sets of single-valued sections.

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.