Volume 21, issue 7 (2021)

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Chris Kottke and Richard Melrose

Algebraic & Geometric Topology 21 (2021) 3335–3399

The bigerbes introduced here give a refinement of the notion of 2–gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form “bundle 2–gerbes” in two ways; this structure replaces higher associativity conditions. We provide natural examples, including a Brylinski–McLaughlin bigerbe associated to a principal G–bundle for a simply connected simple Lie group. This represents the first Pontryagin class of the bundle, and is the obstruction to the lifting problem on the associated principal bundle over the loop space to the structure group consisting of a central extension of the loop group; in particular, trivializations of this bigerbe for a spin manifold are in bijection with string structures on the original manifold. Other natural examples represent “decomposable” 4–classes arising as cup products, a universal bigerbe on K(,4) involving its based double loop space, and the representation of any 4–class on a space by a bigerbe involving its free double loop space. The generalization to “multigerbes” of arbitrary degree is also described.

gerbe, 2–gerbe, bigerbe, multigerbe, loop space, string structure
Mathematical Subject Classification 2010
Primary: 53C08, 55R65
Received: 6 November 2019
Revised: 3 November 2020
Accepted: 11 January 2021
Published: 28 December 2021
Chris Kottke
Division of Natural Sciences
New College of Florida
Sarasota, FL
United States
Richard Melrose
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States