03.06.09 · modern-geometry / characteristic-classes

Dixmier-Douady class and $H^{3} (M, Z)$

shipped3 tiersLean: none

Anchor (Master): Brylinski *Loop Spaces, Characteristic Classes and Geometric Quantization* (Birkhäuser 1993) Ch. 5; Murray *Bundle gerbes* (J. London Math. Soc. 54, 1996) §1-§6; Dixmier-Douady *Champs continus d'espaces hilbertiens et de C*-algèbres* (Bull. Soc. Math. France 91, 1963); Giraud *Cohomologie non-abélienne* (Grundlehren 179, Springer 1971)

Intuition [Beginner]

A line bundle is a way of attaching a one-dimensional vector space to every point of a shape, with a recipe for how the fibres twist as you move from one chart to another. The first Chern class is a number-valued record of how much the line bundle twists, and it lives in the second cohomology of the base. Two line bundles with the same first Chern class are the same; two with different first Chern classes are genuinely different.

A gerbe is one step up the ladder. Where a line bundle has fibres that are vector spaces and transition rules built from numbers in $U (1)$ , a gerbe has fibres that are themselves categories of line bundles, and transition rules built from full line bundles on overlaps. The twisting that records a gerbe needs one extra cohomological degree, and it lives in the third cohomology of the base. That cohomology class is called the Dixmier-Douady class.

The reason this matters: gerbes show up wherever the natural object to attach to a base point is not a vector but a one-dimensional set of bundles. String theory's B-field is a gerbe connection. Twisted K-theory is K-theory twisted by a gerbe. The first Chern class classifies line bundles; the Dixmier-Douady class classifies gerbes.

Visual [Beginner]

A schematic showing two columns. On the left, a line bundle with its first Chern class drawn as a class in second cohomology — a closed 2-form integrated over a sphere returns an integer. On the right, a gerbe with its Dixmier-Douady class drawn as a class in third cohomology — a closed 3-form integrated over a 3-sphere returns an integer. An arrow between the columns shows the one-step shift in cohomological degree.

The picture captures the essential message: a line bundle is degree-two data, a gerbe is degree-three data, and the same machinery that produces the first Chern class produces the Dixmier-Douady class one degree up.

Worked example [Beginner]

Build a gerbe on the three-sphere $S^{3}$ whose Dixmier-Douady class is the canonical generator of the third cohomology group $H^{3} (S^{3}, Z) = Z$ .

Step 1. The three-sphere has third cohomology $H^{3} (S^{3}, Z) = Z$ , generated by the volume class, the closed 3-form $ω$ whose integral over $S^{3}$ is $1$ . The first and second cohomology groups vanish, so there are no interesting line bundles on $S^{3}$ . The first interesting twisting lives in degree three.

Step 2. The Wess-Zumino-Witten gerbe is the standard construction. Cover $S^{3}$ by two open hemispheres $U_{+}$ and $U_{-}$ that overlap on a thickened equator. On the overlap there is a circle of line bundles parametrised by the equator, and the way the circle closes up around the equator is the gerbe data.

Step 3. The Dixmier-Douady class is computed by integrating the closed 3-form $ω$ over the three-sphere. The answer is $1$ , the canonical generator. Halving the construction gives $ω /2$ which is not closed integer-valued, so the half-gerbe does not exist as an honest gerbe on $S^{3}$ . Doubling gives the generator $2$ , and so on.

Step 4. Compare with the simple case where the gerbe data agrees on overlaps in a coherent way over $S^{3}$ . The Dixmier-Douady class is zero. The corresponding gerbe is the plain one, the identity in the classification.

What this tells us: every gerbe on $S^{3}$ is classified by one integer, the Dixmier-Douady class. The Wess-Zumino-Witten gerbe is the generator. This is the exact gerbe-level analogue of the Hopf bundle, whose first Chern class is the generator of $H^{2} (S^{2}, Z) = Z$ .

Check your understanding [Beginner]

Formal definition [Intermediate+]

A bundle gerbe on a smooth manifold $M$ (Murray 1996) is the data of:

a surjective submersion $π : Y \to M$ ,
a $U (1)$ -bundle $L \to Y^{[2]}$ on the fibre product $Y^{[2]} = Y \times_{M} Y$ ,
a bundle gerbe product, an isomorphism of $U (1)$ -bundles on $Y^{[3]} = Y \times_{M} Y \times_{M} Y$ of the form $m : π_{12}^{*} L \otimes π_{23}^{*} L \sim π_{13}^{*} L$ , where $π_{ij} : Y^{[3]} \to Y^{[2]}$ are the projection maps,

satisfying the associativity condition that on $Y^{[4]} = Y \times_{M} Y \times_{M} Y \times_{M} Y$ the two composites $π_{12}^{*} L \otimes π_{23}^{*} L \otimes π_{34}^{*} L \sim π_{14}^{*} L$ obtained by bracketing in the two orders coincide.

The point of the surjective-submersion presentation is that any open cover ${U_{α}}$ of $M$ produces a surjective submersion $Y = ⨆_{α} U_{α} \to M$ , and the fibre products $Y^{[n]}$ recover the $n$ -fold intersections $U_{α_{1} \dots α_{n}}$ . A bundle gerbe presented by an open cover is therefore a $U (1)$ -bundle on double intersections with associativity-compatible isomorphisms on triple intersections. This is the form gerbes take in Brylinski (1993) and Hitchin (1999).

Stable equivalence. Two bundle gerbes $(Y, L, m)$ and $(Y^{'}, L^{'}, m^{'})$ on the same base $M$ are stably isomorphic (Murray-Stevenson 2000) if there exist line bundles $R \to Y$ and $R^{'} \to Y^{'}$ such that the gerbes obtained by twisting $L$ by $R$ and $L^{'}$ by $R^{'}$ become isomorphic bundle gerbes over a common refinement of $Y$ and $Y^{'}$ .

The Dixmier-Douady class. Choose an open cover ${U_{α}}$ of $M$ refining the surjective submersion $Y$ , so that local sections $s_{α} : U_{α} \to Y$ exist. The pulled-back $U (1)$ -bundle $s_{α}^{*} L$ on double intersections becomes a $U (1)$ -bundle $L_{α β} \to U_{α} \cap U_{β}$ . Choose local trivialisations and let $g_{α β γ} : U_{α} \cap U_{β} \cap U_{γ} \to U (1)$ be the cocycle representing the resulting associativity on triple overlaps. The map $g_{α β γ}$ is a Čech 2-cocycle with values in the sheaf $O_{M}^{\infty, *}$ of smooth $U (1)$ -valued functions, and the Dixmier-Douady class of the gerbe is

DD (L) := δ [g_{α β γ}] \in H^{3} (M, Z),

where $δ : H^{2} (M, O_{M}^{\infty, *}) \to H^{3} (M, Z)$ is the connecting homomorphism in the long exact sequence of sheaf cohomology associated to the exponential short exact sequence of sheaves on $M$ ,

0 \to \underline{Z} \to O_{M}^{\infty} e x p (2 π i \cdot) O_{M}^{\infty, *} \to 0.

The smooth sheaf $O_{M}^{\infty}$ is fine (it admits a partition of unity), so $H^{k} (M, O_{M}^{\infty}) = 0$ for $k \geq 1$ , and the connecting homomorphism $δ$ is an isomorphism $H^{k} (M, O_{M}^{\infty, *}) \sim H^{k + 1} (M, \underline{Z})$ . So $DD (L) \in H^{3} (M, Z)$ is the image of the Čech class $[g_{α β γ}]$ under this isomorphism.

Connection and curving. A connection on the bundle gerbe $(Y, L, m)$ is a connection $\nabla_{L}$ on the $U (1)$ -bundle $L \to Y^{[2]}$ compatible with the bundle-gerbe product $m$ , together with a curving, a 2-form $B \in Ω^{2} (Y)$ satisfying $F_{L} = π_{2}^{*} B - π_{1}^{*} B$ on $Y^{[2]}$ , where $F_{L} \in Ω^{2} (Y^{[2]})$ is the curvature of $\nabla_{L}$ and $π_{1}, π_{2} : Y^{[2]} \to Y$ are the projections. The 3-form $d B \in Ω^{3} (Y)$ descends to a closed 3-form $H \in Ω^{3} (M)$ , the 3-curvature of the gerbe connection. The de Rham class $[H] \in H^{3} (M, R)$ is the image of $DD (L)$ under the natural map $H^{3} (M, Z) \to H^{3} (M, R)$ .

Key theorem with proof [Intermediate+]

Theorem (Brylinski 1993, Murray 1996: classification of bundle gerbes). The Dixmier-Douady class is a bijection between stable isomorphism classes of bundle gerbes on $M$ and $H^{3} (M, Z)$ :

DD : {bundle gerbes on M} / (stable iso) \sim H^{3} (M, Z) .

Proof. The strategy is to identify both sides with the sheaf-cohomology group $H^{2} (M, O_{M}^{\infty, *})$ and then use the exponential sequence to convert this into $H^{3} (M, Z)$ .

Step 1: gerbes are 2-cocycles with values in $O_{M}^{\infty, *}$ .* Given a bundle gerbe $(Y, L, m)$ , pass to an open cover ${U_{α}}$ of $M$ that admits sections $s_{α} : U_{α} \to Y$ . The pullback $L_{\alpha\beta} := (s_\alpha, s_\beta)^ L $i s a$ U(1) $- b u n d l eo n$ U_\alpha \cap U_\beta $. C h oose l oc a l t r i v ia l i s a t i o n so f e a c h$ L_{\alpha\beta} $. T h e b u n d l e - g er b e p r o d u c t$ m $, r es t r i c t e d t o t r i pl eo v er l a p s, b eco m es a s m oo t h$ U(1) $- v a l u e df u n c t i o n$ g_{\alpha\beta\gamma} : U_\alpha \cap U_\beta \cap U_\gamma \to U(1) $. T h e a ssoc ia t i v i t y o n q u a d r u pl eo v er l a p sr e a d s a s t h e \overset{ˇ}{C} ec h cocy c l eco n d i t i o n$ \delta g_{\alpha\beta\gamma\delta} = g_{\beta\gamma\delta} g_{\alpha\gamma\delta}^{-1} g_{\alpha\beta\delta} g_{\alpha\beta\gamma}^{-1} = 1 $. S o$ g $i s a \overset{ˇ}{C} ec h 2 - cocy c l e v a l u e d in$ \mathcal{O}_M^{\infty, *} $, d e f inin g a c l a ss$ [g] \in H^2(M, \mathcal{O}_M^{\infty, *})$.

A change of local trivialisation of $L_{α β}$ by a smooth $h_{α β} : U_{α} \cap U_{β} \to U (1)$ alters $g_{α β γ}$ by the Čech coboundary $δ h = h_{β γ} h_{α γ}^{- 1} h_{α β}$ , so the cohomology class $[g]$ is independent of the chosen trivialisations. A different choice of sections $s_{α}$ or refinement of the cover changes $g$ by a coboundary as well. The class depends only on the bundle gerbe up to stable isomorphism: a stable isomorphism by a line bundle $R \to Y$ changes the local data by the Čech coboundary of the transition functions of $R$ .

*Step 2: every class in $H^{2} (M, O_{M}^{\infty, *})$ comes from a bundle gerbe.* Given a Čech 2-cocycle $g_{α β γ}$ on an open cover ${U_{α}}$ , build the surjective submersion $Y = ⨆_{α} U_{α} \to M$ . The double fibre product $Y^{[2]}$ is $⨆_{α, β} U_{α} \cap U_{β}$ , on which the prescribed $U (1)$ -bundles $L_{α β}$ live as the simple $U (1) \times U_{α} \cap U_{β}$ . The triple fibre product $Y^{[3]}$ is $⨆_{α, β, γ} U_{α} \cap U_{β} \cap U_{γ}$ , on which the bundle-gerbe product is defined by the cocycle $g_{α β γ}$ . Associativity on $Y^{[4]}$ is the cocycle condition.

Step 3: the exponential sequence. The short exact sequence $$ 0 \to \underline{\mathbb{Z}} \to \mathcal{O}_M^\infty \xrightarrow{\exp(2\pi i \cdot)} \mathcal{O}_M^{\infty, *} \to 0 $$ of sheaves on $M$ induces the long exact sequence of sheaf cohomology $$ \cdots \to H^k(M, \mathcal{O}_M^\infty) \to H^k(M, \mathcal{O}_M^{\infty, *}) \xrightarrow{\delta} H^{k+1}(M, \underline{\mathbb{Z}}) \to H^{k+1}(M, \mathcal{O}_M^\infty) \to \cdots. $$ The sheaf $O_{M}^{\infty}$ is fine, so $H^{k} (M, O_{M}^{\infty}) = 0$ for $k \geq 1$ . Hence $δ : H^{k} (M, O_{M}^{\infty, *}) \to H^{k + 1} (M, Z)$ is an isomorphism for $k \geq 1$ , and in particular $$ \delta : H^2(M, \mathcal{O}_M^{\infty, *}) \xrightarrow{\sim} H^3(M, \mathbb{Z}). $$

Step 4: combine. Step 1 gives an injection from stable-iso classes of bundle gerbes into $H^{2} (M, O_{M}^{\infty, *})$ . Step 2 gives the reverse: every cohomology class is realised. Step 3 identifies the target with $H^{3} (M, Z)$ . The composite $DD$ is the asserted bijection. $□$

Bridge. This builds toward the entire higher-bundle infrastructure of modern geometry. The foundational reason it holds is exactly that line bundles are classified by $H^{2} (M, Z)$ via the same exponential-sequence mechanism: a line bundle is a 1-cocycle with values in $O_{M}^{\infty, *}$ , and the first Chern class is the image under $δ : H^{1} (M, O_{M}^{\infty, *}) \to H^{2} (M, Z)$ . The Dixmier-Douady class generalises this picture one categorification step up: gerbes are 2-cocycles, and the class lives one degree higher. The same pattern appears again in the classification of $n$ -gerbes by $H^{n + 2} (M, Z)$ . The central insight is that the exponential sequence is a categorification engine — it pushes $U (1)$ -data into one degree higher of integer cohomology — and bundle gerbes are the second step on a ladder whose first step is line bundles and whose general step is $n$ -gerbes. Putting these together, every higher-categorical $U (1)$ -twisting on $M$ is classified by an integer cohomology class one degree above the cocycle degree, and the Dixmier-Douady class is the prototype. The bridge is the recognition that the connecting homomorphism of the exponential sequence is exactly what identifies higher $U (1)$ -bundle data with integer cohomology classes, and this construction appears again in 03.04.14 (hypercohomology) when packaging Deligne cohomology and the connective-and-curving refinement of bundle gerbes.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Let $f : N \to M$ be a smooth map and let $(Y, L, m)$ be a bundle gerbe on $M$ . Construct the pullback bundle gerbe $f^{*} (Y, L, m)$ on $N$ and verify that $DD (f^{*} (Y, L, m)) = f^{*} DD (Y, L, m)$ in $H^{3} (N, Z)$ .

Hint

Pull back the surjective submersion $π : Y \to M$ to $N$ via the fibre product $f^{*} Y = Y \times_{M} N$ . The $U (1)$ -bundle $L$ on $Y^{[2]}$ pulls back to $(f^{*} Y)^{[2]}$ . Functoriality of sheaf cohomology and naturality of the connecting homomorphism do the rest.

Answer

The pullback surjective submersion is $f^{*} Y := Y \times_{M} N \to N$ , with $(f^{*} Y)^{[k]} ≅ f^{*} Y^{[k]}$ for every $k$ . The pulled-back $U (1)$ -bundle is $f^{*} L$ on $(f^{*} Y)^{[2]}$ , and the pulled-back bundle-gerbe product is $f^{*} m$ . Local data: if ${U_{α}}$ covers $M$ with sections $s_{α} : U_{α} \to Y$ , then ${f^{- 1} (U_{α})}$ covers $N$ and $s_{α} \circ f$ gives sections of $f^{*} Y$ , with Čech cocycle $f^{*} g_{α β γ} = g_{α β γ} \circ f$ . The Čech cohomology class is the pullback class $f^{*} [g] \in H^{2} (N, O_{N}^{\infty, *})$ , and naturality of the connecting homomorphism $δ$ gives $DD (f^{*} (Y, L, m)) = δ f^{*} [g] = f^{*} δ [g] = f^{*} DD (Y, L, m)$ . Rubric: full credit for the cocycle-level argument and a naturality statement for $δ$ .

Exercise 4 (medium, short-answer).

Show that on a manifold $M$ with $H^{3} (M, Z) = 0$ (for example $M = R^{n}$ or $M = S^{2}$ ), every bundle gerbe on $M$ is stably isomorphic to the simple gerbe.

Hint

By the classification theorem, stable-iso classes of bundle gerbes on $M$ are in bijection with $H^{3} (M, Z)$ . When that group vanishes, the bijection forces every gerbe class to be the identity.

Answer

By Brylinski-Murray, the map $DD : {bundle gerbes on M} / (stable iso) \to H^{3} (M, Z)$ is a bijection. If $H^{3} (M, Z) = 0$ , then there is a unique stable-iso class, and every bundle gerbe represents the identity element. So every bundle gerbe on such an $M$ is stably isomorphic to the simple one. Examples: $M = R^{n}$ (contractible, all higher cohomology vanishes), $M = S^{2}$ (cohomology in degree zero and two only), $M = T^{2}$ (cohomology in degrees zero, one, two only). Rubric: full credit for the bijection argument and one explicit example.

Exercise 5 (medium, symbolic).

Verify that the Dixmier-Douady class is additive under the natural tensor product of bundle gerbes: if $(Y, L, m)$ and $(Y^{'}, L^{'}, m^{'})$ are bundle gerbes on $M$ , then $DD (L \otimes L^{'}) = DD (L) + DD (L^{'})$ in $H^{3} (M, Z)$ .

Hint

The tensor product of bundle gerbes is presented on a common refinement by the pointwise tensor product of the two $U (1)$ -bundles on $Y^{[2]}$ . At the Čech-cocycle level, this is multiplication of $U (1)$ -valued cocycles, which is addition in the abelian group $H^{2} (M, O_{M}^{\infty, *}) ≅ H^{3} (M, Z)$ .

Answer

Refine $Y$ and $Y^{'}$ to a common surjective submersion $Z \to M$ over which both gerbes have local sections. On $Z^{[2]}$ the $U (1)$ -bundles pull back to $L_{Z}$ and $L_{Z}^{'}$ ; their tensor product $L_{Z} \otimes L_{Z}^{'}$ is again a $U (1)$ -bundle on $Z^{[2]}$ , with bundle-gerbe product the tensor of the original products on $Z^{[3]}$ . At the Čech level on a cover refining $Z$ , the cocycles $g_{α β γ}, g_{α β γ}^{'}$ representing $L_{Z}$ and $L_{Z}^{'}$ combine in the tensor product as $g_{α β γ} g_{α β γ}^{'}$ (pointwise multiplication in $U (1)$ ). The map $H^{2} (M, O_{M}^{\infty, *}) δ H^{3} (M, Z)$ is a group homomorphism, so $δ (g \cdot g^{'}) = δ g + δ g^{'}$ , and $DD (L \otimes L^{'}) = DD (L) + DD (L^{'})$ . Rubric: full credit for the refinement plus the homomorphism-of- $δ$ argument.

Exercise 6 (medium, short-answer).

Explain why the de Rham 3-curvature $H \in Ω^{3} (M)$ of a gerbe connection is a closed form, and identify the cohomology class $[H] \in H^{3} (M, R)$ in terms of the Dixmier-Douady class.

Hint

Use the descent of $d B \in Ω^{3} (Y)$ to $M$ , the closedness of $d B$ on $Y$ (since $d^{2} = 0$ ), and the fact that the resulting class on $M$ is the image of $DD$ under the change-of-coefficients map $H^{3} (M, Z) \to H^{3} (M, R)$ .

Answer

The curving $B \in Ω^{2} (Y)$ has $π_{2}^{*} B - π_{1}^{*} B = F_{L}$ on $Y^{[2]}$ . Apply the exterior derivative: $π_{2}^{*} d B - π_{1}^{*} d B = d F_{L} = 0$ (Bianchi identity). So $d B \in Ω^{3} (Y)$ pulls back the same way along $π_{1}$ and $π_{2}$ , hence descends to a 3-form $H \in Ω^{3} (M)$ . Closedness $d H = 0$ follows from $d^{2} B = 0$ on $Y$ and descent. The de Rham class $[H] \in H^{3} (M, R)$ is the image of $DD (L) \in H^{3} (M, Z)$ under the inclusion $H^{3} (M, Z) ↪ H^{3} (M, R)$ induced by $Z ↪ R$ , via the canonical isomorphism between Čech and de Rham cohomology with $R$ coefficients. Rubric: full credit for the descent argument and the integrality statement.

Exercise 7 (hard, short-answer).

Prove that the Dixmier-Douady class of the tautological gerbe associated to a principal $PU (H)$ -bundle $P \to M$ is the obstruction to lifting $P$ to a principal $U (H)$ -bundle, and identify this obstruction with the third integer cohomology class of $M$ classifying $P$ .

Hint

The central extension $1 \to U (1) \to U (H) \to PU (H) \to 1$ has classifying space sequence $K (Z, 2) \to B U (H) \to B PU (H)$ . The next step $B PU (H) \to K (Z, 3)$ is the classifying map for the obstruction.

Answer

The central extension $1 \to U (1) \to U (H) \to PU (H) \to 1$ of topological groups (where $H$ is a separable infinite-dimensional Hilbert space) gives a fibration $B U (1) = K (Z, 2) \to B U (H) \to B PU (H)$ . Since $U (H)$ is contractible (Kuiper 1965), $B U (H)$ is contractible, and $B PU (H)$ has the homotopy type of $K (Z, 3)$ . A principal $PU (H)$ -bundle $P \to M$ is classified by a map $M \to B PU (H) ≃ K (Z, 3)$ , equivalently by a class in $[M, K (Z, 3)] = H^{3} (M, Z)$ . The lifting $P \to M$ to a principal $U (H)$ -bundle exists iff this class vanishes, since the fibration has fibre $K (Z, 2) = B U (1)$ and the obstruction is the classifying class in $H^{3} (M, Z)$ . The tautological bundle gerbe of $P$ has $Y = P$ , $L \to P^{[2]}$ the $U (1)$ -bundle constructed from the central extension over the equivalent fibres, and Dixmier-Douady class equal to the obstruction. This was the originating example for Dixmier-Douady (1963): the third cohomology class classifying continuous-trace C*-algebras with spectrum $M$ is exactly the Dixmier-Douady class of the associated $PU (H)$ -bundle. Rubric: full credit for the central-extension argument and the identification with $[M, K (Z, 3)]$ .

Exercise 8 (hard, short-answer).

Show that the Dixmier-Douady class is the only $Z$ -valued characteristic class of bundle gerbes that is natural under pullback and additive under tensor product. Specifically, prove that any natural transformation $c : {bundle gerbes on -} / (stable iso) \to H^{*} (-, Z)$ that is a group homomorphism factors through $DD$ .

Hint

The functor ${bundle gerbes} / \sim$ on smooth manifolds is represented by $K (Z, 3)$ . Natural transformations from this representable functor to integer cohomology are classified by elements of $H^{*} (K (Z, 3), Z)$ .

Answer

By Brylinski-Murray, the contravariant functor $Gerbe (-) : Man \to Ab$ , $M \mapsto {bundle gerbes on M} / (stable iso)$ , is naturally isomorphic to $H^{3} (-, Z)$ , which is represented on the homotopy category by the Eilenberg-MacLane space $K (Z, 3)$ . By the Yoneda lemma applied to the homotopy category of CW complexes, natural transformations $Gerbe (-) \to H^{k} (-, Z)$ correspond bijectively to elements of $H^{k} (K (Z, 3), Z)$ . The cohomology of $K (Z, 3)$ in low degrees is computed by Serre: $H^{0} = Z$ , $H^{1} = 0$ , $H^{2} = 0$ , $H^{3} = Z$ generated by the fundamental class, $H^{4} = 0$ . So the natural $Z$ -valued classes of degree three are multiples of the fundamental class — equivalently, multiples of $DD$ — and the additivity requirement makes them integer multiples of $DD$ . Higher-degree natural classes exist (e.g., Steenrod powers of $DD$ ), but in degree three the Dixmier-Douady class is the unique generator up to integer scaling. Rubric: full credit for the Yoneda + $K (Z, 3)$ computation argument.

Advanced results [Master]

Theorem (representation of $DD$ via the connective curving). Let $(Y, L, m)$ be a bundle gerbe on $M$ equipped with a connection $\nabla_{L}$ on $L$ and a curving $B \in Ω^{2} (Y)$ . The descended closed 3-form $H \in Ω^{3} (M)$ satisfies $[H] = DD (L) \otimes_{Z} R \in H^{3} (M, R)$ , and $H$ is integral: $\int_{Σ} H \in Z$ for every closed oriented 3-cycle $Σ$ in $M$ .

The integrality of the de Rham representative is the gerbe-level analogue of the integrality of the curvature of a $U (1)$ -connection. Just as a closed 2-form $F$ with integral periods is the curvature of a line-bundle connection precisely when its periods land in $2 π i Z$ , a closed 3-form $H$ with integral periods is the 3-curvature of a bundle-gerbe connection. The Wess-Zumino-Witten 3-form $H = \frac{1}{6} tr (g^{- 1} d g)^{\land 3}$ on a compact simply-connected Lie group $G$ has integral periods and so defines an honest gerbe on $G$ , the WZW gerbe.

Theorem (Deligne cohomology and the connective gerbe). The data of a bundle gerbe with connection and curving on $M$ is classified by the smooth Deligne cohomology group $H^{3} (M, Z (3)_{D}^{\infty})$ , where $Z (3)_{D}^{\infty}$ is the smooth Deligne complex $\underline{Z} \to O_{M}^{\infty} d Ω_{M}^{1} d Ω_{M}^{2}$ placed in degrees $0$ through $3$ . The forgetful map from connective gerbes to bare gerbes is the natural map $H^{3} (M, Z (3)_{D}^{\infty}) \to H^{3} (M, Z)$ given by the projection $Z (3)_{D}^{\infty} \to Z$ on the leftmost factor.

This packages the Dixmier-Douady class together with the connection and curving data into a single hypercohomology class. The smooth Deligne cohomology $H^{n} (M, Z (n)_{D}^{\infty})$ is the natural home of higher characteristic-class data: $n = 1$ recovers $U (1)$ -valued functions, $n = 2$ recovers line bundles with connection, $n = 3$ recovers bundle gerbes with connection and curving, and $n \geq 4$ recovers connective $n$ -gerbes. This is the hypercohomology view of the entire characteristic-class hierarchy.

Theorem (twisted K-theory; Bouwknegt-Carey-Mathai-Murray-Stevenson 2002). A class $α \in H^{3} (M, Z)$ — equivalently a stable-iso class of bundle gerbes $G_{α}$ on $M$ — defines a twisted K-theory group $K^(M, \alpha) $, w hi c ha g r ees w i t h or d ina r y K - t h eor y$ K^(M) $w h e n$ \alpha = 0 $, an d w hi c hi s g e n u in e l y d i f f er e n tw h e n$ \alpha \neq 0$. The twisted K-theory is defined as the K-theory of the C-algebra of sections of the Hilbert-space bundle associated to the gerbe $G_{α}$ , and is a homotopy-invariant functor of the pair $(M, α)$ .*

Twisted K-theory is the natural recipient of D-brane charges in string theory in the presence of a non-zero B-field, and the B-field is exactly the connection on the gerbe whose Dixmier-Douady class is the K-theory twist class. The third cohomology of $M$ thereby determines an infinite family of K-theory groups, one per gerbe class, and the Dixmier-Douady class is the parameter that runs through this family.

Theorem (gerbes via $PU (H)$ -bundles; Dixmier-Douady 1963 originator). Let $H$ be a separable infinite-dimensional Hilbert space. Stable-iso classes of bundle gerbes on $M$ are in bijection with isomorphism classes of principal $PU (H)$ -bundles on $M$ , via the construction that sends a gerbe to its associated $PU (H)$ -bundle of unit operators on the Hilbert-space bundle attached to the gerbe, and inversely sends a $PU (H)$ -bundle to the bundle gerbe of liftings to $U (H)$ . The Dixmier-Douady class of the gerbe coincides with the third cohomology class classifying the $PU (H)$ -bundle.

This is the originating definition: Dixmier and Douady (1963) introduced the class in $H^{3} (M, Z)$ as the obstruction to lifting a $PU (H)$ -bundle (which they called a continuous-trace C*-algebra) to a $U (H)$ -bundle. The Brylinski-Murray reformulation in terms of bundle gerbes recovers the same class through a much more geometric construction, but the underlying invariant is the same.

Theorem ( $n$ -gerbes and the cohomology ladder). For each $n \geq 0$ , there is a notion of $n$ -gerbe on $M$ , and stable-iso classes of $n$ -gerbes on $M$ are in bijection with $H^{n + 2} (M, Z)$ . The cases are: $n = 0$ gives line bundles classified by $H^{2} (M, Z)$ (first Chern class); $n = 1$ gives bundle gerbes classified by $H^{3} (M, Z)$ (Dixmier-Douady class); $n = 2$ gives 2-gerbes classified by $H^{4} (M, Z)$ ; and so on. The classifying space of $n$ -gerbes is the Eilenberg-MacLane space $K (Z, n + 2)$ .

The ladder reveals that the Dixmier-Douady class is not a standalone construction but a single rung. The exponential sequence and the Čech-de-Rham machinery produce the entire ladder uniformly: at each level, $n$ -cocycles with values in $O_{M}^{\infty, *}$ are identified via the connecting homomorphism with integer cohomology in degree $n + 2$ .

Synthesis. The Dixmier-Douady class is the foundational reason that bundle gerbes — categorified line bundles — are classified by $H^{3} (M, Z)$ . The central insight is that the exponential sequence of sheaves $0 \to \underline{Z} \to O_{M}^{\infty} \to O_{M}^{\infty, *} \to 0$ is a categorification engine: it identifies the obstruction-theoretic $H^{k} (M, O_{M}^{\infty, *})$ with the integer cohomology $H^{k + 1} (M, Z)$ , and the gerbe-level case $k = 2$ is exactly the Dixmier-Douady classification. Putting these together, line bundles correspond to first Chern classes in $H^{2} (M, Z)$ at the $k = 1$ level, gerbes correspond to Dixmier-Douady classes in $H^{3} (M, Z)$ at the $k = 2$ level, and $n$ -gerbes are classified by $H^{n + 2} (M, Z)$ at the $k = n + 1$ level. The bridge is the recognition that the connecting homomorphism of the exponential sequence is exactly what identifies higher categorified $U (1)$ -bundle data with integer cohomology classes one degree above the cocycle degree.

This same mechanism appears again in 03.04.14 (hypercohomology of a complex of sheaves) when packaging Deligne cohomology and the connective-and-curving refinement: the Deligne complex $Z (n)_{D}^{\infty}$ classifies $n$ -gerbes equipped with full connective and curving data, refining the bare class in $H^{n + 2} (M, Z)$ . The bridge is the recognition that hypercohomology of the Deligne complex is the natural ambient where the entire characteristic-class hierarchy lives, with the Dixmier-Douady class being its third rung. This pattern appears again in 03.04.13 (singular cohomology and the de Rham theorem) when matching de Rham 3-curvatures with integer Čech classes, and identifies higher $U (1)$ -twistings with integer cohomology in the next degree up — the foundational reason every gerbe-level question reduces to a computation in $H^{3} (M, Z)$ .

Full proof set [Master]

Proposition (the exponential sequence is exact on a smooth manifold). The short sequence of sheaves on a smooth manifold $M$ $$ 0 \to \underline{\mathbb{Z}} \to \mathcal{O}_M^\infty \xrightarrow{\exp(2\pi i \cdot)} \mathcal{O}_M^{\infty, *} \to 0 $$ is exact, where $\underline{Z}$ is the constant sheaf of integers, $O_{M}^{\infty}$ is the sheaf of smooth $R$ -valued functions, and $O_{M}^{\infty, *}$ is the sheaf of smooth $U (1) \subset C^{*}$ -valued functions.

Proof. Exactness is a stalk-by-stalk check. Pick $x \in M$ and work in a small contractible neighbourhood $U ∋ x$ . The map $exp (2 π i \cdot) : C^{\infty} (U, R) \to C^{\infty} (U, U (1))$ has kernel exactly the locally-constant integer-valued functions on $U$ , which is $Z$ since $U$ is contractible. So the kernel sheaf is $\underline{Z}$ . For surjectivity, any smooth $U (1)$ -valued function $f : U \to U (1)$ on a contractible $U$ admits a smooth logarithm: choose a smooth lift $\tilde{f} : U \to R$ with $exp (2 π i \tilde{f}) = f$ by integrating $d lo g f / (2 π i)$ along paths from the chosen basepoint $x \in U$ . The integration is well-defined because $U$ is simply connected. So $exp (2 π i \cdot)$ is surjective at the level of stalks. $□$

Proposition (the smooth sheaf $O_{M}^{\infty}$ has no higher cohomology). For any paracompact smooth manifold $M$ , $H^{k} (M, O_{M}^{\infty}) = 0$ for every $k \geq 1$ .

Proof. The sheaf $O_{M}^{\infty}$ is fine: there is a smooth partition of unity ${ρ_{α}}$ subordinate to any open cover ${U_{α}}$ of $M$ , and the multiplication-by- $ρ_{α}$ maps $O_{M}^{\infty} \to O_{M}^{\infty}$ give the fineness structure. Fine sheaves have vanishing higher cohomology on paracompact spaces by the standard argument (sheaf cohomology computed via flasque resolutions; fine sheaves are acyclic). So $H^{k} (M, O_{M}^{\infty}) = 0$ for $k \geq 1$ . $□$

Proposition (Dixmier-Douady is well-defined on stable-iso classes). Let $(Y, L, m)$ be a bundle gerbe on $M$ and let $R \to Y$ be a $U (1)$ -bundle. The twisted bundle gerbe $(Y, L \otimes \pi_2^ R \otimes \pi_1^* R^{-1}, m') $ha s t h es am eD i x mi er - D o u a d y c l a ss a s$ (Y, L, m) $, w h er e$ \pi_1, \pi_2 : Y^{[2]} \to Y $a r e t h e p r o j ec t i o n s an d$ m' $i s t h e b u n d l e - g er b e p r o d u c t in d u ce df r o m$ m$.*

Proof. At the level of Čech cocycles on an open cover refining the surjective submersion, the twisting replaces the transition functions $g_{α β γ}$ by $g_{α β γ} \cdot δ r_{α β γ}$ , where $r_{α β}$ are the transition functions of $R$ and $δ r_{α β γ} := r_{β γ} r_{α γ}^{- 1} r_{α β}$ is the Čech coboundary. Since $δ r$ is a coboundary, the cohomology class $[g] \in H^{2} (M, O_{M}^{\infty, *})$ is unchanged, and so is its image under the connecting homomorphism $δ : H^{2} (M, O_{M}^{\infty, *}) \to H^{3} (M, Z)$ . $□$

Proposition (Dixmier-Douady is a group homomorphism). The set of stable-iso classes of bundle gerbes on $M$ forms an abelian group under tensor product, and $DD : (this group) \to H^{3} (M, Z)$ is a group homomorphism.

Proof. The tensor product of two bundle gerbes $(Y, L, m)$ and $(Y^{'}, L^{'}, m^{'})$ on $M$ is defined on a common refinement $Z \to M$ of $Y$ and $Y^{'}$ as $(Z, L_{Z} \otimes L_{Z}^{'}, m_{Z} \otimes m_{Z}^{'})$ , where the subscript $Z$ denotes pullback. Associativity of the bundle-gerbe product is preserved under tensor product. The simple gerbe with $L = U (1) \times M^{[2]}$ and trivialized product is the identity element. The inverse of $(Y, L, m)$ is $(Y, L^{- 1}, m^{- 1})$ where $L^{- 1}$ is the dual $U (1)$ -bundle on $Y^{[2]}$ .

At the Čech level, tensor product is pointwise multiplication of cocycles: $(g \otimes g^{'})_{α β γ} = g_{α β γ} g_{α β γ}^{'}$ . The connecting homomorphism $δ : H^{2} (M, O_{M}^{\infty, *}) \to H^{3} (M, Z)$ is a group homomorphism between abelian groups, so $DD (L \otimes L^{'}) = δ (g \cdot g^{'}) = δ g + δ g^{'} = DD (L) + DD (L^{'})$ . $□$

Proposition (de Rham representative of $DD$ via curving). Let $(Y, L, m)$ be a bundle gerbe on $M$ with connection $\nabla_{L}$ on $L$ and curving $B \in Ω^{2} (Y)$ . The 3-form $H \in Ω^{3} (M)$ descended from $d B \in Ω^{3} (Y)$ is closed, and its de Rham class $[H] \in H^{3} (M, R)$ is the image of $DD (L) \in H^{3} (M, Z)$ under the natural map $H^{3} (M, Z) \to H^{3} (M, R)$ .

Proof. The curving condition reads $π_{2}^{*} B - π_{1}^{*} B = F_{L}$ on $Y^{[2]}$ , where $F_{L} \in Ω^{2} (Y^{[2]})$ is the curvature of $\nabla_{L}$ . Apply the exterior derivative: $π_{2}^{*} d B - π_{1}^{*} d B = d F_{L} = 0$ by the Bianchi identity. So $d B \in Ω^{3} (Y)$ has $π_{1}^{*} d B = π_{2}^{*} d B$ , meaning $d B$ descends to a 3-form $H \in Ω^{3} (M)$ . Closedness $d H = 0$ follows from $d^{2} B = 0$ on $Y$ and the injectivity of pullback by a surjective submersion.

Identifying $[H] \in H^{3} (M, R)$ with the image of $DD (L)$ proceeds via the Čech-de-Rham double complex (see 03.04.11). The Čech-de-Rham complex with values in $Ω^{∙}$ computes $H^{*} (M, R)$ , and the Čech complex with values in $O_{M}^{\infty, *}$ maps to it via $lo g / (2 π i)$ on the diagonal. The induced map on cohomology sends the Čech class $[g_{α β γ}]$ representing $DD (L)$ to the de Rham class of the 3-form $H$ built from the curving. This is the precise identification of integer Čech cohomology with real de Rham cohomology under the natural map. $□$

Proposition (the WZW gerbe on a compact Lie group has Dixmier-Douady class generating $H^{3}$ ). Let $G$ be a compact, simple, simply-connected Lie group. The closed 3-form $H_{G} := \frac{1}{6} tr (g^{- 1} d g)^{\land 3} \in Ω^{3} (G)$ is integral, and there is a bundle gerbe on $G$ — the Wess-Zumino-Witten gerbe — whose 3-curvature is $H_{G}$ and whose Dixmier-Douady class is the canonical generator of $H^{3} (G, Z) = Z$ .

Proof. Integrality of $H_{G}$ : for a compact simple simply-connected $G$ , $H^{3} (G, Z) = Z$ generated by the class dual to the Cartan-Killing 3-form, and a direct computation (or the equivalent statement that the WZW level-one path integral is well-defined on closed 3-manifolds) shows that the periods of $H_{G}$ on 3-cycles of $G$ are integers. Specifically, $\int_{S U (2)} H_{G} = 1$ on the generating 3-sphere $S U (2) ↪ G$ .

Existence of the gerbe: choose a covering ${U_{α}}$ of $G$ refining the Bruhat-cell decomposition, and on each $U_{α}$ choose a 2-form primitive $B_{α}$ of $H_{G}$ (each $U_{α}$ is contractible). On overlaps $U_{α} \cap U_{β}$ , the difference $B_{β} - B_{α}$ is closed; since $U_{α} \cap U_{β}$ is contractible, it is exact, so $B_{β} - B_{α} = d A_{α β}$ for a 1-form $A_{α β}$ . On triple overlaps, $A_{α β} + A_{β γ} - A_{α γ}$ is closed; integrality of $H_{G}$ gives a $U (1)$ -cocycle $g_{α β γ} = exp (2 π i \int A_{α β γ})$ satisfying the bundle-gerbe associativity. This is the Wess-Zumino-Witten gerbe.

The Dixmier-Douady class equals the de Rham class of $H_{G}$ , which is the generator of $H^{3} (G, Z) = Z$ . $□$

Proposition (classification: every class in $H^{3} (M, Z)$ is realised). For every $α \in H^{3} (M, Z)$ there exists a bundle gerbe $(Y, L, m)$ on $M$ with $DD (L) = α$ .

Proof. Lift $α$ along the isomorphism $δ : H^{2} (M, O_{M}^{\infty, *}) \sim H^{3} (M, Z)$ to a Čech 2-cocycle $g_{α β γ}$ on some open cover ${U_{α}}$ of $M$ . Construct the surjective submersion $Y := ⨆_{α} U_{α} \to M$ , the $U (1)$ -bundle $L \to Y^{[2]} = ⨆_{α, β} U_{α} \cap U_{β}$ as the simple $U (1)$ -bundle, and the bundle-gerbe product $m$ on $Y^{[3]} = ⨆_{α, β, γ} U_{α} \cap U_{β} \cap U_{γ}$ as multiplication by $g_{α β γ}$ . Associativity on $Y^{[4]}$ is the cocycle condition $δ g = 1$ . By construction the Dixmier-Douady class is $δ [g] = α$ . $□$

Connections [Master]

Pontryagin and Chern classes 03.06.04. The first Chern class is the line-bundle analogue of the Dixmier-Douady class: line bundles on $M$ are classified by $H^{2} (M, Z)$ via the first Chern class, and gerbes on $M$ are classified by $H^{3} (M, Z)$ via the Dixmier-Douady class. The constructions are parallel: both arise from the connecting homomorphism of the exponential sequence $0 \to \underline{Z} \to O_{M}^{\infty} \to O_{M}^{\infty, *} \to 0$ , applied to a Čech 1-cocycle in the line-bundle case and to a Čech 2-cocycle in the gerbe case. The two constructions are the first two rungs of the $n$ -gerbe ladder.
Hypercohomology of a complex of sheaves 03.04.14. The connective gerbe with curving is classified by the smooth Deligne hypercohomology $H^{3} (M, Z (3)_{D}^{\infty})$ , refining the bare Dixmier-Douady class in $H^{3} (M, Z)$ by the connection and curving data. Hypercohomology is the natural ambient where the entire ladder of $n$ -gerbes with connective data lives, with the Deligne complex packaging $\underline{Z} \to O_{M}^{\infty} \to Ω_{M}^{1} \to Ω_{M}^{2}$ in the gerbe case. The Dixmier-Douady class is the projection of the Deligne class to its leftmost factor.
Čech-de Rham double complex 03.04.11. The identification of the integer Čech class with the de Rham 3-curvature $H \in Ω^{3} (M)$ proceeds through the Čech-de-Rham double complex. The map $lo g / (2 π i) : O_{M}^{\infty, *} \to O_{M}^{\infty}$ on a local trivialisation patches together to send a Čech 2-cocycle in $O_{M}^{\infty, *}$ to a 3-form in $Ω^{3} (M)$ via the double complex. The diagonal of the double complex realises the integer Dixmier-Douady class as a real de Rham class.
Singular cohomology and the de Rham theorem 03.04.13. The Dixmier-Douady class lives in $H^{3} (M, Z)$ , the integer cohomology of $M$ . The integer-coefficient class refines the real-coefficient class given by the de Rham 3-curvature, and the integrality of the periods of the 3-curvature on closed 3-cycles is exactly the integrality content of the Dixmier-Douady classification. The de Rham theorem identifies real Čech and real de Rham cohomology, but the integer structure that distinguishes a gerbe from a 3-form requires the full integer cohomology.
Chern-Weil homomorphism 03.06.06. Chern-Weil theory builds characteristic classes of principal bundles from invariant polynomials on the Lie algebra. The Dixmier-Douady class is not a Chern-Weil class of a principal $G$ -bundle for a finite-dimensional $G$ ; it is instead the obstruction class of an infinite-dimensional $PU (H)$ -bundle, and the gerbe-with-curving formalism provides the analogue of the Chern-Weil construction one categorification step up. The 3-curvature $H$ plays the role of the curvature $F$ in Chern-Weil, and the integrality of its periods is the gerbe-level analogue of integrality of the first Chern class.
Principal bundle 03.05.01. The originating Dixmier-Douady picture (1963) is a principal $PU (H)$ -bundle on $M$ together with its obstruction class to lifting to a principal $U (H)$ -bundle. The Brylinski-Murray bundle-gerbe reformulation rephrases this $PU (H)$ -bundle picture in a finite-dimensional surjective-submersion language, but the equivalence of the two perspectives is exactly Exercise 7. Principal bundles in the categorified setting (principal $\infty$ -bundles) recover the entire $n$ -gerbe ladder.

Historical & philosophical context [Master]

The class now bearing the name "Dixmier-Douady" was introduced by Jacques Dixmier and Adrien Douady in their 1963 paper Champs continus d'espaces hilbertiens et de C-algèbres* (Bull. Soc. Math. France 91, 227-284) ^[pending]. The setting was operator-algebraic: a continuous-trace C*-algebra with spectrum $M$ is essentially a bundle of compact operators over $M$ , and Dixmier-Douady showed that such an algebra is classified up to Morita equivalence by a class in $H^{3} (M, Z)$ . The class measures the obstruction to the bundle of compact operators being globally of the form $End (E)$ for a Hilbert-space bundle $E$ over $M$ ; equivalently, the obstruction to lifting a principal $PU (H)$ -bundle to a principal $U (H)$ -bundle.

The geometric reformulation as a class attached to a higher categorical bundle (a gerbe) came through Jean Giraud's work on non-abelian cohomology, Cohomologie non-abélienne (Grundlehren der mathematischen Wissenschaften 179, Springer 1971) ^[pending]. Giraud introduced the term "gerbe" in the original stack-theoretic sense — a sheaf of groupoids satisfying certain descent conditions — and classified gerbes banded by a sheaf of abelian groups $A$ by the second non-abelian cohomology $H^{2} (M, A)$ . Applied to $A = O_{M}^{\infty, *}$ , Giraud's framework gives the gerbe-level classification by $H^{3} (M, Z)$ via the exponential sequence.

The presentation that has become standard among differential geometers is due to Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization (Birkhäuser Progress in Mathematics 107, 1993) ^[pending], and Michael Murray, Bundle gerbes (J. London Math. Soc. (2) 54, 1996, 403-416) ^[pending]. Brylinski presented gerbes in the sheaf-of-groupoids language with explicit connective and curving data, and connected them to the Cheeger-Simons differential characters and to Deligne cohomology. Murray reformulated the same data in the more elementary surjective-submersion-plus- $U (1)$ -bundle language used here, eliminating the need for stacks and making the geometry computationally accessible. The two formulations are equivalent.

Nigel Hitchin's lectures on special Lagrangian submanifolds (AMS/IP Stud. Adv. Math. 23, 2001, 151-182) ^[pending] brought gerbes into the string-theory mainstream by identifying the B-field of type II string theory as a connection on a gerbe whose Dixmier-Douady class is the twist class for D-brane charges. The subsequent development of twisted K-theory by Peter Bouwknegt, Alan Carey, Varghese Mathai, Michael Murray, and Danny Stevenson (Commun. Math. Phys. 228, 2002, 17-49) ^[pending] made the Dixmier-Douady class the central invariant of an entire family of K-theory groups $K^{*} (M, α)$ parametrised by $α \in H^{3} (M, Z)$ . The class now sits in three distinct mathematical contexts simultaneously: operator-algebraic (Dixmier-Douady 1963), geometric (Brylinski-Murray 1993-1996), and physical (Hitchin 2001 and followers), with the same invariant playing the central role in each.

Bibliography [Master]

@article{DixmierDouady1963,
  author    = {Dixmier, Jacques and Douady, Adrien},
  title     = {Champs continus d'espaces hilbertiens et de C*-alg{\`e}bres},
  journal   = {Bull. Soc. Math. France},
  volume    = {91},
  year      = {1963},
  pages     = {227--284}
}

@book{Giraud1971,
  author    = {Giraud, Jean},
  title     = {Cohomologie non-ab{\'e}lienne},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {179},
  publisher = {Springer-Verlag},
  year      = {1971}
}

@book{Brylinski1993,
  author    = {Brylinski, Jean-Luc},
  title     = {Loop Spaces, Characteristic Classes and Geometric Quantization},
  series    = {Progress in Mathematics},
  volume    = {107},
  publisher = {Birkh{\"a}user},
  year      = {1993}
}

@article{Murray1996,
  author    = {Murray, Michael K.},
  title     = {Bundle gerbes},
  journal   = {J. London Math. Soc. (2)},
  volume    = {54},
  year      = {1996},
  pages     = {403--416}
}

@incollection{Hitchin2001,
  author    = {Hitchin, Nigel J.},
  title     = {Lectures on special {L}agrangian submanifolds},
  booktitle = {Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds},
  series    = {AMS/IP Stud. Adv. Math.},
  volume    = {23},
  publisher = {American Mathematical Society},
  year      = {2001},
  pages     = {151--182}
}

@article{MurrayStevenson2000,
  author    = {Murray, Michael K. and Stevenson, Daniel},
  title     = {Bundle gerbes: stable isomorphism and local theory},
  journal   = {J. London Math. Soc. (2)},
  volume    = {62},
  year      = {2000},
  pages     = {925--937}
}

@article{BCMMS2002,
  author    = {Bouwknegt, Peter and Carey, Alan L. and Mathai, Varghese and Murray, Michael K. and Stevenson, Danny},
  title     = {Twisted K-theory and K-theory of bundle gerbes},
  journal   = {Commun. Math. Phys.},
  volume    = {228},
  year      = {2002},
  pages     = {17--49}
}

@phdthesis{Chatterjee1998,
  author    = {Chatterjee, David Saumitra},
  title     = {On Gerbs},
  school    = {Trinity College, University of Cambridge},
  year      = {1998}
}

@article{Hitchin2010,
  author    = {Hitchin, Nigel J.},
  title     = {Lectures on generalized geometry},
  journal   = {Adv. Lect. Math.},
  volume    = {16},
  year      = {2010},
  pages     = {79--124}
}

@article{Kuiper1965,
  author  = {Kuiper, Nicolaas H.},
  title   = {The homotopy type of the unitary group of {H}ilbert space},
  journal = {Topology},
  volume  = {3},
  year    = {1965},
  pages   = {19--30}
}

Prerequisites

03.06.04
03.04.14
03.04.11
03.04.13
03.05.01

Tier anchors

beginner: Hitchin *Lectures on Special Lagrangian Submanifolds* (2001) informal opening on gerbes as bundles with one extra degree
intermediate: Brylinski *Loop Spaces, Characteristic Classes and Geometric Quantization* (Birkhäuser 1993) §5; Murray *Bundle gerbes* (J. London Math. Soc. 1996) §1-§3
master: Brylinski *Loop Spaces, Characteristic Classes and Geometric Quantization* (Birkhäuser 1993) Ch. 5; Murray *Bundle gerbes* (J. London Math. Soc. 54, 1996) §1-§6; Dixmier-Douady *Champs continus d'espaces hilbertiens et de C*-algèbres* (Bull. Soc. Math. France 91, 1963); Giraud *Cohomologie non-abélienne* (Grundlehren 179, Springer 1971)

References

TODO_REF
Brylinski — Loop Spaces, Characteristic Classes and Geometric Quantization · Birkhäuser Progress in Mathematics 107, 1993; Ch. 5 sheaves of groupoids and gerbes; §5.2 Dixmier-Douady class; §5.3 connective structure and curving; §5.4 classification by $H^3(M, \mathbb{Z})$
TODO_REF
Murray — Bundle gerbes · J. London Math. Soc. (2) 54 (1996), 403-416; original modern definition of a bundle gerbe over a surjective submersion $Y \to M$ with $U(1)$-bundle on $Y^{[2]}$; §1 the definition, §3 the Dixmier-Douady class, §6 stable equivalence
TODO_REF
Dixmier-Douady — Champs continus d'espaces hilbertiens et de C*-algèbres · Bull. Soc. Math. France 91 (1963), 227-284; originator paper defining the obstruction class in $H^3(X, \mathbb{Z})$ for continuous-trace C*-algebras
TODO_REF
Giraud — Cohomologie non-abélienne · Grundlehren der mathematischen Wissenschaften 179, Springer 1971; §IV gerbes (in the original stack-theoretic sense); $H^2$ with values in a sheaf of groups classifies gerbes
TODO_REF
Hitchin — Lectures on Special Lagrangian Submanifolds · in *Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds*, AMS/IP Stud. Adv. Math. 23 (2001), 151-182; gerbes in string theory, the B-field as a connection on a gerbe
TODO_REF
Murray-Stevenson — Bundle gerbes: stable isomorphism and local theory · J. London Math. Soc. (2) 62 (2000), 925-937; the stable-isomorphism equivalence relation under which the Dixmier-Douady class is a complete invariant
TODO_REF
Chatterjee — On gerbs · PhD thesis, Cambridge 1998; companion presentation of gerbes via local data and the Dixmier-Douady class as a degree-three Čech cocycle
TODO_REF
Hitchin — Lectures on generalized geometry · Adv. Lect. Math. 16 (2010), 79-124; review of bundle gerbes, curving, and the role of the Dixmier-Douady class in twisted K-theory
TODO_REF
Bouwknegt-Carey-Mathai-Murray-Stevenson — Twisted K-theory and K-theory of bundle gerbes · Commun. Math. Phys. 228 (2002), 17-49; the Dixmier-Douady class as the twist class for K-theory in string theory

Reviewer

TBD

Estimated time

beginner: 18m
intermediate: 40m
master: 85m