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A [[Cochain|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.
 
A [[Cochain|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c0228301.png" /> with coefficients in a non-Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c0228302.png" /> is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c0228303.png" />, defined on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c0228304.png" /> of one-dimensional simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c0228305.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c0228306.png" /> for any two-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c0228307.png" />. Two cocycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c0228308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c0228309.png" /> are said to be cohomologous if there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c02283010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c02283011.png" /> for any one-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c02283012.png" />. The cohomology classes of one-dimensional cocycles form a pointed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022830/c02283013.png" />. 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.
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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 $
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with coefficients in a non-Abelian group $  G $
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is a function $  \sigma \rightarrow f ( \sigma ) \in G $,  
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defined on the set $  K _ {1} $
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of one-dimensional simplices of $  K $,  
 +
such that $  f ( \sigma  ^ {(} 0) ) f ( \sigma  ^ {(} 2) ) = f ( \sigma  ^ {(} 1) ) $
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for any two-dimensional simplex $  \sigma \in K $.  
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Two cocycles $  f $
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and $  g $
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are said to be cohomologous if there exists a function $  h:  K _ {0} \rightarrow G $
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such that $  f ( \tau ) h ( \tau  ^ {(} 0) ) = h ( \tau  ^ {(} 1) ) g ( \tau ) $
 +
for any one-dimensional simplex $  \tau \in K $.  
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The cohomology classes of one-dimensional cocycles form a pointed set $  H  ^ {1} ( K;  G) $.  
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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|Cochain]].

Revision as of 17:45, 4 June 2020


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