Transgression
in a fibre space
A certain special correspondence between the cohomology classes of the fibre and the base. More precisely, if is a connected fibre space with base
and fibre
and
is an Abelian group, then a transgression in
is the correspondence
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defined by the formula
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where is the coboundary homomorphism and
is the homomorphism determined by the projection
. The elements of the domain of definition
of the correspondence
are called transgressive; any
such that
is called the image of the element
under transgression. A transgression can be regarded as a homomorphism of the group
into some quotient group of
. The transgression has a transparent interpretation in terms of the spectral sequence
of the fibre space
: in essence, it is the same as the differential
.
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 is a field,
for
,
is the exterior algebra over a subspace
, graded in odd degrees, where the cohomology spaces of the fibres form a simple sheaf over
, then
can be chosen such that
for any
; furthermore,
is the algebra of polynomials in images of elements of a homogeneous basis of
under the transgression. In particular, if
is a connected Lie group without
-torsion and
, then
, where the homogeneous elements of
have odd degree and are transgressive in any principal fibre bundle of the group
. Here
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 |
Comments
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) |
Transgression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transgression&oldid=17798