Zoran Skoda transgression

There is plenty of mutually related notions in topology and geometry, especially cohomology theory?, called transgression.

One classical case is in the theory of fiber bundles (as studied in classical works by Borel and Hirzebruch). Given a topological group $G$ the basic transgression map is the natural map

$$\tau : H^k(G,\mathbb{Z})\to H^{k-1}(BG,\mathbb{Z})$$

Recall that for any fiber bundle $\pi : E\to X$ with fiber $F$, there is the corresponding spectral sequence of the fiber bundle (both for homology and for cohomology). In particular there is a spectral sequence for the universal fiber bundle $EG\to BG$.

For cohomology, $E_\infty = Gr H^*(E,A)$, $E^{p,q}_2 = H^p (X,\tilde{H}^q(F,A))$ where $A$ is the module of coefficients (over some ground ring, additional assumptions apply) and $\tilde{H}^q(X,A)$ is the corresponding local system $(H^*(F_b,A),\phi^*)$ (it will be explained later, $b$ is the base point). The differential $d_{q+1}: E^{0,q}_{q+1}\to E^{q+1,0}_{q+1}$ is called the transgression.

Geometrically this corresponds to the follows. The inclusion of the fiber $i_b : F_b\hookrightarrow E$ and the projection $\pi:E\to X$ induce cochain maps

$$i^*: C^q(E,A)\to C^q(F,A),\,\,\,\,\,\pi^*: C^q(X,A)\to C^q(E,A)$$

An element $f\in H^q(F,A)$ is said to be transgressive if there exist $e\in C^q(E,A)$ with $[i^*e]= f$ and $\delta e = \pi^*(b)$ where $b\in C^{q+1}(X,A)$. Then $\tau(x) = [b]$.

• Armand Borel, La transgression dans les espaces fibrés principaux, C. R. Acad. Sci. Paris 232, (1951). 2392–2394.

• A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57, (1953). 115–207, doi (Russian translation in collection Болтянский В.Г., Дынкин Е.Б., Постников М.М. (ред.) Расслоенные пространства и их приложения (сборник переводов) ИЛ, 1958 458 p. file)