cohomology

# Contents

## Idea

A bundle gerbe or circle 2-bundle has a unique characteristic class in integral cohomology in degree 3, the higher analog of the Chern class of a circle group-principal bundle (or complex line bundle): this is called the Dixmier-Douady class of the bundle gerbe.

## Definition

In the literature one find a universal Dixmier-Douady class defined for different entities, notably for projective unitary-principal bundles and for $U(1)$-bundle gerbes, as well as for C-star algebra constructions related to these. All these notions are equivalent in one sense, namely in bare homotopy theory, but differ in other sense, namely in geometric homotopy theory.

### In bare homotopy-type theory

The classifying space of the circle 2-group $\mathbf{B}U(1)$ is an Eilenberg-MacLane space $B \mathbf{B} U(1) \simeq B^3 \mathbb{Z} \simeq K(\mathbb{Z}, 3)$. The bare Dixmier-Douday class is the universal characteristic class

$DD : B B U(1) \stackrel{\simeq}{\to} K(\mathbb{Z}, 3)$

exhibited by this equivalence. Hence if we identify $B B U(1)$ with $K(\mathbb{Z}, 3)$, then the DD-class is the identity on this space.

This is directly analogous to how the first Chern class is, as a universal characteristic class, the identity on $K(\mathbb{Z},2) \simeq B U(1)$.

This means conversely that the equivalence class of a $U(1)$-bundle gerbe/circle 2-bundle is entirely characterized by its Dixmier-Douady class.

### In smooth homotopy-type theory

The circle 2-group $\mathbf{B}U(1)$ naturally carries a smooth structure, hence is naturally regarded not just as an ∞-group in ∞Grpd, but as a smooth ∞-group in $\mathbf{H} \coloneqq$ Smooth∞Grpd.

For each $n$, the central extension of Lie groups

$U(1) \to U(n) \to PU(n)$

that exhibits the unitary group as a circle group-extension of the projective unitary group induces the corresponding morphism of smooth moduli stacks

$\mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n)$

in $\mathbf{H}$.

This is part of a long fiber sequence in $\mathbf{H}$ which continues to the right by a connecting homomorphism $\mathbf{dd}_n$

$\mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n) \stackrel{\mathbf{dd}_n}{\to} \mathbf{B}^2 U(1)$

in $\mathbf{H}$. Here the last morphism is presented in simplicial presheaves by the zig-zag/∞-anafunctor of sheaves of crossed modules

$\array{ [U(1) \to U(n)] &\to& [U(1) \to 1] \\ {}^{\mathllap{\simeq}}\downarrow \\ PU(n) } \,.$

To get rid of the dependence on the rank $n$ – to stabilize the rank – we may form the directed colimit of smooth moduli stacks

$\mathbf{B}U \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} U(n)$
$\mathbf{B} PU \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} PU(n) \,.$

On these we have the smooth universal class

$\mathbf{dd} : \mathbf{B} PU \to \mathbf{B}^2 U(1) \,.$

Since the (∞,1)-topos Smooth∞Grpd has universal colimits, it follows that there is a fiber sequence

$\array{ \mathbf{B}U &\to& \mathbf{B} PU \\ && \downarrow^{\mathbf{dd}} \\ && \mathbf{B}^2 U(1) }$

exhibiting the moduli stack of smooth stable unitary bundles as the homotopy fiber of $\mathbf{dd}$.

## References

Revised on May 30, 2012 16:20:39 by David Corfield (129.12.18.29)