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Difference between revisions of "Cocycle"

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defined on the set  $  K _ {1} $
 
defined on the set  $  K _ {1} $
 
of one-dimensional simplices of  $  K $,  
 
of one-dimensional simplices of  $  K $,  
such that  $  f ( \sigma  ^ {(} 0) ) f ( \sigma  ^ {(} 2) ) = f ( \sigma  ^ {(} 1) ) $
+
such that  $  f ( \sigma  ^ {(0)}) f ( \sigma  ^ {(2)} ) = f ( \sigma  ^ {(1)} ) $
 
for any two-dimensional simplex  $  \sigma \in K $.  
 
for any two-dimensional simplex  $  \sigma \in K $.  
 
Two cocycles  $  f $
 
Two cocycles  $  f $
 
and  $  g $
 
and  $  g $
 
are said to be cohomologous if there exists a function  $  h:  K _ {0} \rightarrow G $
 
are said to be cohomologous if there exists a function  $  h:  K _ {0} \rightarrow G $
such that  $  f ( \tau ) h ( \tau  ^ {(} 0) ) = h ( \tau  ^ {(} 1) ) g ( \tau ) $
+
such that  $  f ( \tau ) h ( \tau  ^ {(0)} ) = h ( \tau  ^ {(1)} ) g ( \tau ) $
 
for any one-dimensional simplex  $  \tau \in K $.  
 
for any one-dimensional simplex  $  \tau \in K $.  
 
The cohomology classes of one-dimensional cocycles form a pointed set  $  H  ^ {1} ( K;  G) $.  
 
The cohomology classes of one-dimensional cocycles form a pointed set  $  H  ^ {1} ( K;  G) $.  
 
Similarly one defines one-dimensional cocycles and their cohomology classes in the Aleksandrov–Čech sense, with coefficients in a sheaf of non-Abelian groups. The cohomology groups of these cocycles are related to fibre bundles with a structure group.
 
Similarly one defines one-dimensional cocycles and their cohomology classes in the Aleksandrov–Čech sense, with coefficients in a sheaf of non-Abelian groups. The cohomology groups of these cocycles are related to fibre bundles with a structure group.
  
For references see [[Cochain|Cochain]].
+
For references see [[Cochain]].

Latest revision as of 20:47, 16 January 2024


A cochain which is annihilated by the coboundary mapping, in other words, a cochain that vanishes on boundary chains. The concept of a cocycle generalizes the concept of a closed differential form on a smooth manifold with a vanishing integral over a boundary chain.

In accordance with the different versions of the concept of a cochain, there are different versions of cocycles. For example, an Aleksandrov–Čech cocycle in a topological space is a cocycle of the nerve of some open covering of the space. Only one-dimensional cocycles with non-Abelian coefficients need special discussion. A one-dimensional cocycle of a simplicial set $ K $ with coefficients in a non-Abelian group $ G $ is a function $ \sigma \rightarrow f ( \sigma ) \in G $, defined on the set $ K _ {1} $ of one-dimensional simplices of $ K $, such that $ f ( \sigma ^ {(0)}) f ( \sigma ^ {(2)} ) = f ( \sigma ^ {(1)} ) $ for any two-dimensional simplex $ \sigma \in K $. Two cocycles $ f $ and $ g $ are said to be cohomologous if there exists a function $ h: K _ {0} \rightarrow G $ such that $ f ( \tau ) h ( \tau ^ {(0)} ) = h ( \tau ^ {(1)} ) g ( \tau ) $ for any one-dimensional simplex $ \tau \in K $. The cohomology classes of one-dimensional cocycles form a pointed set $ H ^ {1} ( K; G) $. Similarly one defines one-dimensional cocycles and their cohomology classes in the Aleksandrov–Čech sense, with coefficients in a sheaf of non-Abelian groups. The cohomology groups of these cocycles are related to fibre bundles with a structure group.

For references see Cochain.

How to Cite This Entry:
Cocycle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cocycle&oldid=46376
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article