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 $U\left(1\right)$-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 $BU\left(1\right)$ is an Eilenberg-MacLane space $BBU\left(1\right)\simeq {B}^{3}ℤ\simeq K\left(ℤ,3\right)$. The bare Dixmier-Douday class is the universal characteristic class

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

exhibited by this equivalence. Hence if we identify $BBU\left(1\right)$ with $K\left(ℤ,3\right)$, 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 $K\left(ℤ,2\right)\simeq BU\left(1\right)$.

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

### In smooth homotopy-type theory

The circle 2-group $BU\left(1\right)$ naturally carries a smooth structure, hence is naturally regarded not just as an ∞-group in ∞Grpd, but as a smooth ∞-group in $H≔$ Smooth∞Grpd.

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

$U\left(1\right)\to U\left(n\right)\to \mathrm{PU}\left(n\right)$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

$BU\left(1\right)\to BU\left(n\right)\to B\mathrm{PU}\left(n\right)$\mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n)

in $H$.

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

$BU\left(1\right)\to BU\left(n\right)\to B\mathrm{PU}\left(n\right)\stackrel{{\mathrm{dd}}_{n}}{\to }{B}^{2}U\left(1\right)$\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 $H$. Here the last morphism is presented in simplicial presheaves by the zig-zag/∞-anafunctor of sheaves of crossed modules

$\begin{array}{ccc}\left[U\left(1\right)\to U\left(n\right)\right]& \to & \left[U\left(1\right)\to 1\right]\\ {}^{\simeq }↓\\ \mathrm{PU}\left(n\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$BU≔\underset{{\to }_{n}}{\mathrm{lim}}BU\left(n\right)$\mathbf{B}U \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} U(n)
$B\mathrm{PU}≔\underset{{\to }_{n}}{\mathrm{lim}}B\mathrm{PU}\left(n\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{B} PU \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} PU(n) \,.

On these we have the smooth universal class

$\mathrm{dd}:B\mathrm{PU}\to {B}^{2}U\left(1\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}BU& \to & B\mathrm{PU}\\ & & {↓}^{\mathrm{dd}}\\ & & {B}^{2}U\left(1\right)\end{array}$\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 $\mathrm{dd}$.

## References

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