# Bigerbes

## TL;DR

This paper introduces bigerbes, a refined mathematical structure representing degree four cohomology classes, with applications to principal bundles, string structures, and higher cohomology theories.

## Contribution

It defines bigerbes using bisimplicial line bundles, providing new models for degree four cohomology and their relation to string structures and higher classes.

## Key findings

- Bigerbes represent the first Pontryagin class of principal G-bundles.
- Trivializations of bigerbes correspond to string structures on manifolds.
- Examples include bigerbes for cup products and universal models on K(Z,4).

## 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(Z,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.