Strict orbifold atlases and weighted branched manifolds

This note revisits some of the ideas in [M1] on orbifolds and branched manifolds, showing how the constructions can be simplified by using a version of the Kuranishi atlases developed by McDuff–Wehrheim. We first show that every orbifold has such an atlas, and then use it to obtain an explicit model for the nonsingular resolution of an oriented orbifold $Y$ (which is a weighted nonsingular groupoid with the same fundamental class as $Y$) and 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.

Keywords

orbifold, groupoid, strict atlas, Kuranishi atlas, weighted branched manifold

2010 Mathematics Subject Classification

53D45, 57R18

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Partially supported by NSF grant DMS1308669.

Received 17 June 2015

Published 26 July 2017