[Paper Review] Bundle gerbes: stable isomorphism and local theory

Michael K. Murray, Daniel Stevenson|ArXiv.org|Aug 26, 1999

Homotopy and Cohomology in Algebraic Topology9 references4 citations

TL;DR

This paper introduces stable isomorphism for bundle gerbes, establishing a bijective correspondence between stable isomorphism classes of bundle gerbes over a manifold M and the third integral cohomology group H^3(M, Z). By developing a local theory and leveraging Gajer's work on BC^x bundles, the authors construct a classifying theory for bundle gerbes, providing a cohomological classification that clarifies their global structure and equivalence relations.

ABSTRACT

We consider the notion of stable isomorphism of bundle gerbes. It has the consequence that the stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with H^3(M, Z). Stable isomorphism sheds light on the local theory of bundle gerbes and enables us to develop a classifying theory for bundle gerbes using results of Gajer on BC^x bundles.

Motivation & Objective

• To define and analyze stable isomorphism for bundle gerbes, a generalization of isomorphism that respects tensor products and trivializations.
• To clarify the local structure of bundle gerbes by introducing stable isomorphism, which simplifies classification.
• To establish a classifying theory for bundle gerbes using results from Gajer on BC^x bundles.
• To prove that stable isomorphism classes of bundle gerbes are in bijective correspondence with H^3(M, Z), providing a cohomological invariant.

Proposed method

• Defining stable isomorphism as an equivalence relation on bundle gerbes that respects tensor products and trivializations.
• Using local trivializations and transition data to analyze the structure of bundle gerbes over open covers.
• Applying Gajer's theory of BC^x bundles to construct a classifying space for bundle gerbes.
• Demonstrating that the obstruction to trivializing a bundle gerbe lies in H^3(M, Z) via the stable isomorphism class.
• Establishing a bijection between stable isomorphism classes and H^3(M, Z) using Čech cohomology and bundle gerbe cocycles.
• Leveraging the fact that stable isomorphism preserves the Dixmier-Douady class, which classifies bundle gerbes up to stable isomorphism.

Experimental results

Research questions

• RQ1 How can one define an equivalence relation on bundle gerbes that captures their essential geometric and topological invariants beyond strict isomorphism?
• RQ2 What is the role of stable isomorphism in simplifying the local theory of bundle gerbes?
• RQ3 Can bundle gerbes be classified using cohomological invariants, and if so, which cohomology group classifies them?
• RQ4 How does the use of BC^x bundles from Gajer's work contribute to a classifying theory for bundle gerbes?
• RQ5 What is the precise relationship between the stable isomorphism class of a bundle gerbe and its Dixmier-Douady class in H^3(M, Z)?

Key findings

• Stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with elements of the third integral cohomology group H^3(M, Z).
• Stable isomorphism provides a natural framework for understanding the local structure of bundle gerbes, particularly in terms of transition data and trivializations.
• The classifying theory for bundle gerbes is constructed using Gajer's results on BC^x bundles, which allows for a geometric realization of the cohomology class in H^3(M, Z).
• The Dixmier-Douady class of a bundle gerbe is invariant under stable isomorphism, confirming its role as a complete invariant.
• The paper establishes that the obstruction to a bundle gerbe being trivial is measured precisely by its class in H^3(M, Z), under stable isomorphism.
• The theory shows that stable isomorphism is the correct equivalence relation for classifying bundle gerbes up to geometric and topological equivalence.