A complex line bundle is classified topologically by a degree-two Chern class. A bundle gerbe shifts the pattern one degree: its characteristic obstruction is the Dixmier–Douady class in third integral cohomology.
The presentation uses a surjective submersion, a line bundle over a double fibre product and a multiplication over triple overlaps. Different presentations can encode the same gerbe class, so strict isomorphism is too rigid for classification.
The Gold Standard is to track the fibre-product data, associativity and local cocycle; compute the degree-three class; compare presentations after allowed trivial factors; and distinguish stable isomorphism from connection-preserving or differential refinements.
A practical orientation
Use this guide after the general gerbes-and-descent article. The worked example follows a bundle gerbe over the three-sphere with Dixmier–Douady class 2 and tests tensor product, dual, pullback and stable triviality.
Choose a reading route
Work and transfer the example
Chapters 7–9 test stable isomorphism, tensor operations and pullback.
Chapter contents
- 1. Recall the line-bundle analogy without flattening it
- 2. Choose a surjective submersion and fibre products
- 3. Put a line bundle and associative multiplication on overlaps
- 4. Pass to local sections and a Cech 2-cocycle
- 5. Define the Dixmier–Douady class
- 6. Use stable isomorphism instead of strict presentation equality
- 7. Worked example: class 2 on the three-sphere
- 8. Transfer check: pull back along maps of different degree
- 9. Distinguish existence, classification and construction
- 10. Add connection, curving and three-curvature carefully
- 11. State limits and preserve evidence
- 12. FAQs and the next reading route
1. Recall the line-bundle analogy without flattening it
A line bundle can be described by transition functions on double overlaps, with a cocycle condition on triple overlaps. Its first Chern class lies in H2 of the base with integer coefficients. A gerbe uses one higher layer of overlap data and produces a class in H3.
The analogy guides the construction but does not identify a gerbe with an ordinary line bundle. The multiplication, refinements and stable equivalences all live one categorical level higher.
| Object | Local data | Characteristic degree |
|---|---|---|
| Complex line bundle | Functions on double overlaps | H2 |
| Bundle gerbe | Line bundles / functions through triple overlaps | H3 |
| Gerbe with connection | Forms plus cocycle data | Differential H3 |
2. Choose a surjective submersion and fibre products
Let pi from Y to M be a surjective submersion admitting local sections. Form Y two-fold fibre product over M: pairs of points in Y that project to the same base point. Triple and higher fibre products record compatible tuples over one point of M.
The choice of Y is a presentation device. A refinement can replace it with another cover-like object. Classification should therefore not depend on the accidental size or labelling of Y.
- Base manifold M.
- Surjective submersion Y to M.
- Double fibre product Y[2].
- Triple fibre product Y[3].
- Local sections or refinements.
3. Put a line bundle and associative multiplication on overlaps
A bundle gerbe has a line bundle L over Y[2]. Over a triple (y1,y2,y3), the multiplication identifies L over (y1,y2) tensor L over (y2,y3) with L over (y1,y3). Over quadruples, the two ways of multiplying three factors must agree.
Associativity is not decorative. Without it, the local data cannot descend coherently and the later cocycle fails its required identity.
| Fibre product | Data | Coherence |
|---|---|---|
| Y[2] | Line bundle L | Source and target over same base point |
| Y[3] | Multiplication | L12 tensor L23 to L13 |
| Y[4] | Two multiplication paths | Associativity equality |
4. Pass to local sections and a Cech 2-cocycle
Choose an open cover with local sections of Y. Pull L back to each double overlap, choose local trivialisations and compare multiplication on triple overlaps. The comparison produces circle-valued functions g_ijk satisfying a cocycle condition on quadruple overlaps.
Changing sections or trivialisations changes the representative by a coboundary. The cohomology class, after the standard passage to integral degree three, is the invariant rather than one chosen table of functions.
- Open cover chosen.
- Local sections of Y.
- Trivialisations over double overlaps.
- g_ijk on triple overlaps.
- Coboundary changes tracked.
5. Define the Dixmier–Douady class
The Dixmier–Douady class DD(L) lies in H3(M,Z). It vanishes exactly when the bundle gerbe is trivial in the appropriate sense. Tensor product adds classes and the dual gerbe negates the class.
A class calculation is a classification statement for stable isomorphism, not a proof that two presentations have identical Y, line-bundle fibres or multiplication maps.
| Operation | Effect on DD class | Interpretation |
|---|---|---|
| Tensor product | DD1 + DD2 | Combine twisting |
| Dual | -DD | Reverse twisting |
| Pullback f | f-star DD | Transport to new base |
| Trivial gerbe | 0 | Stable triviality |
6. Use stable isomorphism instead of strict presentation equality
Two bundle gerbes can have the same Dixmier–Douady class while their chosen presentations are not strictly isomorphic. Stable isomorphism allows tensoring with trivial gerbes built from line bundles on the presentation spaces, then comparing the resulting objects.
For bundle gerbes over a manifold, equality of Dixmier–Douady classes characterises stable-isomorphism classes. This is the classification test that restores the clean analogy with Chern classes and line bundles.
| Comparison | Requires | Too strong or appropriate |
|---|---|---|
| Strict isomorphism | Presentation-level match | Often too strong |
| Common refinement | Comparable presentation spaces | Preparation step |
| Stable isomorphism | Trivial factors allowed | Classification level |
| Equal DD classes | Same H3 class | Equivalent to stable isomorphism |
7. Worked example: class 2 on the three-sphere
Because H3(S3,Z) is isomorphic to Z, choose a bundle gerbe G whose Dixmier–Douady class is 2 times a generator. Choose another presentation H with the same class 2. Their local covers and line bundles may differ, but the equal classes imply that G and H are stably isomorphic.
The dual G-star has class minus 2. The tensor product G tensor G-star has class zero and is therefore stably trivial. By contrast G tensor H has class 4, so it cannot be stably isomorphic to G.
This arithmetic is a classification calculation. It does not construct the stable isomorphism itself; an explicit equivalence still requires compatible refinement and line-bundle data.
| Gerbe | DD class on S3 | Stable conclusion |
|---|---|---|
| G | 2 | Nontrivial |
| H | 2 | Stably isomorphic to G |
| G-star | -2 | Dual class |
| G tensor G-star | 0 | Stably trivial |
| G tensor H | 4 | Not stably isomorphic to G |
8. Transfer check: pull back along maps of different degree
Let f from S3 to S3 have degree m. Pullback acts on H3 by multiplication by m, so DD(f-star G) equals 2m. A degree-zero map makes the pulled-back class vanish; a degree-three map produces class 6.
Now pull G to a space X with H3(X,Z) equal to zero. The topological class vanishes, so the pullback is stably trivial. This does not automatically choose a canonical trivialisation or erase connection-level holonomy.
| Changed context | Pulled-back class | Conclusion |
|---|---|---|
| Degree 0 map S3 to S3 | 0 | Stably trivial |
| Degree 1 map | 2 | Same class |
| Degree 3 map | 6 | Different class |
| Target with H3 = 0 | 0 | Topologically stably trivial |
9. Distinguish existence, classification and construction
An H3 class answers which stable-isomorphism class is present. It does not by itself provide a convenient finite cover, explicit multiplication, connection or trivialising line bundle. Those are construction problems with additional choices.
Conversely, a complicated presentation can represent the zero class. Do not mistake large local data for nontrivial topology; compute the invariant and track the coboundaries.
- Existence of a representative.
- Computation of DD class.
- Stable classification.
- Explicit stable isomorphism.
- Chosen trivialisation or connection.
10. Add connection, curving and three-curvature carefully
A bundle-gerbe connection compatible with multiplication can be supplemented by a curving on Y. Its differential descends to a closed three-form H on M. The de Rham class of H corresponds to the real image of the Dixmier–Douady class, with the conventional normalisation stated.
Torsion integral classes map to zero in real cohomology, so curvature alone cannot classify every gerbe. Differential refinements retain holonomy and integral information that a three-form by itself loses.
| Level | Data | Blind spot |
|---|---|---|
| Topological | DD in H3(M,Z) | No connection geometry |
| De Rham | Closed 3-form H | Misses torsion |
| Differential | Cocycle, forms and integrality | Requires compatible choices |
| Holonomy | Surface transport | Not a standalone global class |
11. State limits and preserve evidence
Equal real three-curvatures do not imply equal integral Dixmier–Douady classes when torsion is possible. Equal DD classes imply stable isomorphism, not strict equality of chosen connections. A banded gerbe or non-abelian gerbe can require different classification data.
Preserve the base and cover, submersion, fibre products, line bundle, multiplication, associativity check, local sections, cocycle, coboundary changes, DD computation, refinements, trivial factors and any connection or curvature conventions.
- No strict-isomorphism claim from DD alone.
- No torsion conclusion from curvature alone.
- No canonical trivialisation from class zero.
- No silent change of coefficient group.
- No transfer to non-abelian gerbes without new theory.
12. FAQs and the next reading route
Bundle gerbes make degree-three twisting concrete enough to calculate while stable isomorphism keeps the answer independent of presentation. The Dixmier–Douady class is the correct topological label, provided the classification level is stated.
Read the Topology root, the general gerbes-and-band-data article and the constructible-stacks descent article alongside this guide. The parent Gold Standard and How X Works hub connect the same practice across subjects.
Why is the class in H3 rather than H2?
The gerbe descent data sits one overlap level above line-bundle transition functions, shifting the characteristic class by one degree.
Do equal Dixmier–Douady classes give a strict isomorphism?
Not necessarily. They classify bundle gerbes up to stable isomorphism.
Does zero three-curvature mean the gerbe is trivial?
Not always. Torsion Dixmier–Douady classes can have zero image in real cohomology.
Continue through the verified ecosystem
Sources and scope note
- Murray and Stevenson — Bundle Gerbes: Stable Isomorphism and Local Theory — used for stable isomorphism, local cocycles and classification by H3.
- Journal of the London Mathematical Society — Published Stable-Isomorphism Paper — used for the published classification statement and bibliographic record.
- Murray — Bundle Gerbes — used for the original bundle-gerbe construction and Dixmier–Douady class.
The S3 class-2 gerbe and pullback calculations are teaching examples based on standard cohomology. The cited papers establish the definitions and classification results; they do not supply the exact presentation used in the example.