Volume 21, issue 7 (2021)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 21
Issue 7, 3221–3734
Issue 6, 2677–3220
Issue 5, 2141–2676
Issue 4, 1595–2140
Issue 3, 1075–1593
Issue 2, 543–1074
Issue 1, 1–541

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
 
Other MSP Journals
Bigerbes

Chris Kottke and Richard Melrose

Algebraic & Geometric Topology 21 (2021) 3335–3399
Abstract

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.

Keywords
gerbe, 2–gerbe, bigerbe, multigerbe, loop space, string structure
Mathematical Subject Classification 2010
Primary: 53C08, 55R65
References
Publication
Received: 6 November 2019
Revised: 3 November 2020
Accepted: 11 January 2021
Published: 28 December 2021
Authors
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