[Paper Review] Atlases for Ineffective Orbifolds

Dorette Pronk, Laura Scull+1 more|arXiv (Cornell University)|Jun 14, 2016

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper introduces a new definition of orbifold atlases for ineffective orbifolds—generalizing the classical atlas approach to include non-effective group actions—by redefining chart embeddings using group bimodules and abstract embeddings. The key contribution is proving that this new atlas definition is equivalent to the standard topological groupoid definition of ineffective orbifolds via Morita equivalence.

ABSTRACT

We give a definition of atlases for ineffective orbifolds, and prove that this definition leads to the same notion of orbifold as that defined via topological groupoids.

Motivation & Objective

• To resolve the mismatch between existing ineffective orbifold atlas definitions and the standard groupoid-based definition of orbifolds.
• To extend the classical atlas formalism to ineffective orbifolds by redefining chart embeddings using abstract embeddings and group bimodules.
• To establish a rigorous, consistent atlas-based framework for ineffective orbifolds that matches the groupoid approach.
• To provide a foundation for studying suborbifolds and embeddings in the ineffective setting.
• To support the development of generalized orbifold categories via local chart and atlas structures.

Proposed method

• Defining a new orbifold atlas for ineffective orbifolds using group bimodules and abstract embeddings between charts.
• Introducing the concept of abstract embeddings in the category of group actions to generalize chart transition maps.
• Constructing a groupoid from an atlas by defining source and target maps using equivalence classes of triples involving bimodules and transition data.
• Defining a Morita equivalence between groupoids constructed from atlases by constructing a bimodule with compatible left and right groupoid actions.
• Using refinement of atlases to construct a manifold structure on the space of composable triples, ensuring surjective submersions and compatibility with groupoid actions.
• Verifying that the resulting bimodule satisfies the axioms of a Morita equivalence, including fullness, faithfulness, and compatibility with the groupoid actions.

Experimental results

Research questions

• RQ1 How can the classical atlas definition of orbifolds be extended to the ineffective case, where group actions are not necessarily effective?
• RQ2 What structure is needed to correctly define embeddings between ineffective orbifold charts so that the atlas axioms are satisfied?
• RQ3 Does the proposed atlas definition for ineffective orbifolds yield the same orbifold objects as the standard topological groupoid approach?
• RQ4 Can a Morita equivalence be established between the groupoid constructed from an atlas and the groupoid defining the orbifold?
• RQ5 How can the atlas formalism be used to study suborbifolds and embeddings in the ineffective setting?

Key findings

• The proposed atlas definition for ineffective orbifolds is equivalent to the standard definition via proper étale topological groupoids.
• The new definition redefines chart embeddings using group bimodules and abstract embeddings, resolving inconsistencies in prior approaches.
• A Morita equivalence is constructed between the groupoid derived from an atlas and the groupoid defining the orbifold, proving equivalence of the two definitions.
• The construction of the bimodule M(𝒰) involves quotienting a disjoint union of triples by an equivalence relation that respects groupoid actions and transition data.
• The maps τ and ε from M(𝒰)jk to the chart domains are surjective submersions, ensuring the structure is compatible with the groupoid topology.
• The left and right groupoid actions on the bimodule are well-defined and satisfy the axioms of a Morita equivalence, completing the correspondence between atlases and groupoids.