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Dixmier-Douady class

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Cohomology

cohomology

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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)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 BU(1)\mathbf{B}U(1) is an Eilenberg-MacLane space BBU(1)B 3K(,3)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:BBU(1)K(,3) DD : B B U(1) \stackrel{\simeq}{\to} K(\mathbb{Z}, 3)

exhibited by this equivalence. Hence if we identify BBU(1)B B U(1) with K(,3)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(,2)BU(1)K(\mathbb{Z},2) \simeq B U(1).

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

In smooth homotopy-type theory

The circle 2-group BU(1)\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 H\mathbf{H} \coloneqq Smooth∞Grpd.

For each nn, the central extension of Lie groups

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

in H\mathbf{H}.

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

BU(1)BU(n)BPU(n)dd nB 2U(1) \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\mathbf{H}. Here the last morphism is presented in simplicial presheaves by the zig-zag/∞-anafunctor of sheaves of crossed modules

[U(1)U(n)] [U(1)1] PU(n). \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 nn – to stabilize the rank – we may form the directed colimit of smooth moduli stacks

BUlim nBU(n) \mathbf{B}U \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} U(n)
BPUlim nBPU(n). \mathbf{B} PU \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} PU(n) \,.

On these we have the smooth universal class

dd:BPUB 2U(1). \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

BU BPU dd B 2U(1) \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 dd\mathbf{dd}.

References

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