Definition
A transgression map is a homomorphism in algebraic topology that connects the (co)homology groups of the total space, base space, and fiber of a fibration or of a filtered complex. It typically appears in the context of spectral sequences, where it transfers cohomology classes from one page to another, often relating classes in the cohomology of the fiber to those in the cohomology of the base.
Overview
In a Serre fibration $F\hookrightarrow E \xrightarrow{p} B$, the long exact sequence of homotopy groups and the associated Leray–Serre spectral sequence provide a systematic way to compute the cohomology of $E$. The transgression map $ \tau $ arises on the $E_2$-page of this spectral sequence: for a cohomology class $ \alpha \in H^q(F;G) $ that survives to the $E_{r}$-page, $ \tau(\alpha) $ is an element of $ H^{q+1}(B;G) $ representing the obstruction to lifting $ \alpha $ to a class in $ H^{q}(E;G) $. When the fibration is a principal $G$-bundle, the transgression often encodes characteristic classes, such as the first Chern class of a circle bundle being the transgression of the generator of $H^1(S^1;\mathbb{Z})$.
Analogous constructions exist in other settings: for filtered chain complexes, a transgression map links the associated graded pieces; in differential geometry, the term appears in the theory of Chern–Weil forms and in the study of secondary characteristic classes.
Etymology/Origin
The word “transgression” derives from Latin transgressio, meaning “a stepping across.” In the topological context, it reflects the idea of “stepping” a cohomology class from the fiber across to the base. The concept was formalized in the mid‑20th century with the development of spectral sequences by Jean Leray, Henri Cartan, and later elaborated by Jean-Pierre Serre in his seminal work on homotopy groups of spheres (1951).
Characteristics
| Feature | Description |
|---|---|
| Domain | Typically $H^q(F;G)$ or a graded piece of a filtered complex. |
| Codomain | Usually $H^{q+1}(B;G)$ (or a shifted degree in the spectral sequence). |
| Naturality | The map is natural with respect to maps of fibrations; commutes with induced homomorphisms on (co)homology. |
| Degree shift | In cohomology it raises degree by one; in homology it lowers degree by one. |
| Exactness relation | Appears in the long exact sequence of a fibration when the connecting homomorphism coincides with the transgression under certain conditions. |
| Dependence on choice | Defined up to sign and may depend on the choice of a splitting or a connection when expressed in differential‑geometric terms. |
| Applications | Computation of characteristic classes, identification of non‑trivial elements in homotopy groups, analysis of secondary invariants, and construction of products such as the cup‑$i$ products. |
Related Topics
- Leray–Serre spectral sequence – the primary framework where transgression maps are introduced.
- Cohomology of fiber bundles – transgression relates fiber and base cohomology.
- Characteristic classes – many classes arise as transgressions of universal classes.
- Exact couples and spectral sequences – general machinery underlying the definition of transgression.
- Connecting homomorphism – in long exact sequences, the transgression can be viewed as a specific connecting map.
- Secondary characteristic classes – such as the Chern–Simons forms, which are obtained via transgression of primary invariants.