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''in a fibre space''
 
''in a fibre space''
  
A certain special correspondence between the cohomology classes of the fibre and the base. More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t0937501.png" /> is a connected [[Fibre space|fibre space]] with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t0937502.png" /> and fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t0937503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t0937504.png" /> is an Abelian group, then a transgression in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t0937505.png" /> is the correspondence
+
A certain special correspondence between the cohomology classes of the fibre and the base. More precisely, if $  E $
 +
is a connected [[Fibre space|fibre space]] with base $  B $
 +
and fibre $  F $
 +
and $  A $
 +
is an Abelian group, then a transgression in $  E $
 +
is the correspondence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t0937506.png" /></td> </tr></table>
+
$$
 +
\tau  \subset  \
 +
H  ^ {s} ( F, A) \times H ^ {s + 1 } ( B, A)
 +
$$
  
 
defined by the formula
 
defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t0937507.png" /></td> </tr></table>
+
$$
 +
\tau  = \{ {( x, y) } : {\delta x = q  ^ {*} y } \}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t0937508.png" /> is the coboundary homomorphism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t0937509.png" /> is the homomorphism determined by the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375010.png" />. The elements of the domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375011.png" /> of the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375012.png" /> are called transgressive; any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375014.png" /> is called the image of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375015.png" /> under transgression. A transgression can be regarded as a homomorphism of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375016.png" /> into some quotient group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375017.png" />. The transgression has a transparent interpretation in terms of the spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375018.png" /> of the fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375019.png" />: in essence, it is the same as the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375020.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375021.png" /> is a field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375022.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375024.png" /> is the exterior algebra over a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375025.png" />, graded in odd degrees, where the cohomology spaces of the fibres form a simple sheaf over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375027.png" /> can be chosen such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375028.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375029.png" />; furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375030.png" /> is the algebra of polynomials in images of elements of a homogeneous basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375031.png" /> under the transgression. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375032.png" /> is a connected Lie group without <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375033.png" />-torsion and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375035.png" />, where the homogeneous elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375036.png" /> have odd degree and are transgressive in any principal fibre bundle of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375037.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093750/t09375038.png" /> coincides with the space of primitive cohomology classes.
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "Homologie singulière des espaces fibrés. Applications"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 425–505</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "Homologie singulière des espaces fibrés. Applications"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 425–505</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. 360ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Dieudonné,  "A history of algebraic and differential topology 1900–1960" , Birkhäuser  (1989)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. 360ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Dieudonné,  "A history of algebraic and differential topology 1900–1960" , Birkhäuser  (1989)</TD></TR></table>

Latest revision as of 08:26, 6 June 2020


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

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)
How to Cite This Entry:
Transgression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transgression&oldid=17798
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article