[Paper Review] Orbifold Atlases of Kuranishi type

Dusa McDuff|arXiv (Cornell University)|Jun 17, 2015

Homotopy and Cohomology in Algebraic Topology3 citations

TL;DR

This paper introduces a simplified construction of orbifold atlases using Kuranishi structures, proving every orbifold admits such an atlas. It applies this to explicitly model the nonsingular resolution of oriented orbifolds and the Euler class of oriented orbibundles as weighted branched manifolds, where the Euler class arises as the zero set of a single-valued section over the resolution, with weights and branching canonically defined by the atlas.

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.

Motivation & Objective

• To simplify the construction of orbifold atlases using the Kuranishi framework developed by McDuff and Wehrheim.
• To prove that every orbifold admits a Kuranishi atlas of the required type.
• To construct explicit models for the nonsingular resolution of oriented orbifolds as weighted nonsingular groupoids.
• To provide a new geometric interpretation of the Euler class of an oriented orbibundle as the zero set of a single-valued section over the resolution.
• To canonically define weights and branching in the resulting branched manifolds via the atlas structure.

Proposed method

• Adopt the Kuranishi atlas formalism to represent orbifolds with local models that are compatible under transition data.
• Construct a resolution of an oriented orbifold as a weighted nonsingular groupoid, preserving the fundamental class.
• Define the Euler class of an orbibundle as the zero locus of a single-valued section over the resolution, rather than a multivalued section.
• Use the atlas to assign canonical weights and branching data to the zero set, turning it into a weighted branched manifold.
• Ensure consistency of the atlas structure across local charts, enabling global geometric interpretation.
• Leverage the Kuranishi framework to avoid technical complications of earlier constructions in orbifold and branched manifold theory.

Experimental results

Research questions

• RQ1 Can Kuranishi atlases be used to construct a canonical, explicit resolution of an oriented orbifold as a weighted nonsingular groupoid?
• RQ2 How can the Euler class of an oriented orbibundle be reinterpreted geometrically using a resolution and a single-valued section?
• RQ3 What is the role of the atlas in canonically assigning weights and branching to the zero set of a section in the resolution?
• RQ4 Can the fundamental class of an orbifold be preserved in the resolution constructed via Kuranishi atlases?
• RQ5 What structural advantages does the Kuranishi framework offer over earlier approaches to orbifold atlases and branched manifolds?

Key findings

• Every orbifold admits a Kuranishi atlas of the required type, establishing a foundational existence result.
• The nonsingular resolution of an oriented orbifold is explicitly realized as a weighted nonsingular groupoid that preserves the fundamental class.
• The Euler class of an oriented orbibundle is realized as the zero set of a single-valued section over the resolution, avoiding multivalued sections.
• The resulting zero set forms a weighted branched manifold where weights and branching are canonically determined by the atlas structure.
• The construction provides a geometric and canonical framework for studying orbifold invariants via resolution and section theory.
• The method simplifies prior constructions by unifying the treatment of orbifolds and orbibundles through the Kuranishi formalism.