Transgression

in a fibre space

A certain special correspondence between the cohomology classes of the fibre and the base. More precisely, if $ E $ is a connected fibre space with base $ B $ and fibre $ F $ and $ A $ is an Abelian group, then a transgression in $ E $ is the correspondence

$$ \tau \subset \ H ^ {s} ( F, A) \times H ^ {s + 1 } ( B, A) $$

defined by the formula

$$ \tau = \{ {( x, y) } : {\delta x = q ^ {*} y } \} , $$

where $ \delta : H ^ {*} ( F, A) \rightarrow H ^ {s + 1 } ( E, F, A) $ is the coboundary homomorphism and $ q: H ^ {s + 1 } ( B, A) \rightarrow H ^ {s + 1 } ( E, F, A) $ is the homomorphism determined by the projection $ E \rightarrow B $. The elements of the domain of definition $ T ^ {s} ( F, A) $ of the correspondence $ \tau $ are called transgressive; any $ y \in H ^ {s + 1 } ( B, A) $ such that $ x \tau y $ is called the image of the element $ x \in T ^ {s} ( F, A) $ under transgression. A transgression can be regarded as a homomorphism of the group $ T ^ {s} ( F, A) $ into some quotient group of $ H ^ {s + 1 } ( B, A) $. The transgression has a transparent interpretation in terms of the spectral sequence $ ( H _ {r} ) $ of the fibre space $ E $: in essence, it is the same as the differential $ d _ {s + 1 } : H _ {s + 1 } ^ {0,s} \rightarrow H _ {s + 1 } ^ {s + 1, 0 } $.

The description of transgressive cohomology classes of the fibre is very important in the study of the cohomological structure of fibre bundles. An important role is played here by the Borel transgression theorem: If $ A $ is a field, $ H ^ {n} ( E, A) = 0 $ for $ n > 0 $, $ H ^ {*} ( F, A) = \wedge P $ is the exterior algebra over a subspace $ P $, graded in odd degrees, where the cohomology spaces of the fibres form a simple sheaf over $ B $, then $ P $ can be chosen such that $ P ^ {x} = T ^ {s} ( F, A) $ for any $ s > 0 $; furthermore, $ H ^ {*} ( B, A) $ is the algebra of polynomials in images of elements of a homogeneous basis of $ P $ under the transgression. In particular, if $ G $ is a connected Lie group without $ p $- torsion and $ \mathop{\rm char} A = p $, then $ H ^ {*} ( G, A) = \wedge P $, where the homogeneous elements of $ B $ have odd degree and are transgressive in any principal fibre bundle of the group $ G $. Here $ P $ coincides with the space of primitive cohomology classes.

References

[1]	A. Borel, "Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compacts" Ann. of Math. , 57 (1953) pp. 115–207
[2]	J.-P. Serre, "Homologie singulière des espaces fibrés. Applications" Ann. of Math. , 54 (1951) pp. 425–505

References

[a1]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. 360ff
[a2]	J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)