Zoran Skoda

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 GG the basic transgression map is the natural map

τ:H k(G,)H k1(BG,) \tau : H^k(G,\mathbb{Z})\to H^{k-1}(BG,\mathbb{Z})

Recall that for any fiber bundle π:EX\pi : E\to X with fiber FF, 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 EGBGEG\to BG.

For cohomology, E =GrH *(E,A)E_\infty = Gr H^*(E,A), E 2 p,q=H p(X,H˜ q(F,A))E^{p,q}_2 = H^p (X,\tilde{H}^q(F,A)) where AA is the module of coefficients (over some ground ring, additional assumptions apply) and H˜ q(X,A)\tilde{H}^q(X,A) is the corresponding local system (H *(F b,A),ϕ *)(H^*(F_b,A),\phi^*) (it will be explained later, bb is the base point). The differential d q+1:E q+1 0,qE q+1 q+1,0d_{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 bEi_b : F_b\hookrightarrow E and the projection π:EX\pi:E\to X induce cochain maps

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

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

Last revised on May 17, 2010 at 15:43:38. See the history of this page for a list of all contributions to it.