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A homogeneous element of an Abelian cochain group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c0228101.png" /> (or, in the general case, a module), i.e. a graded Abelian group equipped with an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c0228102.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c0228103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c0228104.png" />. The endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c0228105.png" /> is called the coboundary mapping or the coboundary.
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{{TEX|done}}
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{{MSC|18G35|55Nxx}}
  
A cochain group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c0228106.png" /> usually arises as a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c0228107.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c0228108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c0228109.png" /> is an arbitrary Abelian group, called the coefficient group, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281010.png" /> is a chain group, i.e. a graded Abelian group equipped with an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281011.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281012.png" /> (the boundary mapping or boundary) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281013.png" />. In this situation the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281014.png" /> on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281015.png" /> is defined as the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281018.png" />.
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===Definition and properties===
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A homogeneous element of an Abelian cochain group $C^\star$ (or, in the general case, a module). A cochain group $C^\star$ is a [[Graded module|graded Abelian]] group, which means that $C^\star$ is decomposed as the direct sum of subgroups $A_k$, indexed with $k\in \mathbb Z$, some of which might be trivial; an homogenenous element $f$ is an element belonging to some $A_k$, where $k$ is called the degree of the element and denoted by ${\rm deg}\, (f)$. The cochain group is also equipped with an endomorphism $\delta: C^\star \to C^\star$ of degree $+ 1$, namely mapping elements of $A_k$ into elements of $A_{k+1}$, such that $\delta \circ \delta =0$. The endomorphism $\delta$ is called the coboundary mapping or the coboundary.
  
Given a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281019.png" />, one defines the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281020.png" /> of singular chains as the Abelian group of formal finite sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281022.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281023.png" /> are arbitrary singular simplices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281024.png" />, i.e. continuous mappings of the standard simplex into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281025.png" />. A singular cochain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281026.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281027.png" /> is defined as a homogeneous element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281028.png" />.
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====Duality====
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A cochain group $C^\star$ arises often as dual of a [[Chain|chain group]] $C_\star$ with coefficients group $G$, i.e. as a group $C^\star = {\rm Hom} (C_\star, G)$, where
 +
* $G$ is an arbitrary Abelian group
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* $C_\star$, the chain group, is a graded Abelian group equipped with an endomorphism $\partial$ of degree $-1$ (the boundary mapping or boundary) with $\partial \circ \partial =0$. $C_\star$ has an operation of multiplication by elements of $G$, namely a map $C_\star \times G \to C_\star$ which is an homomorphism on each factor.
  
Similarly, a simplicial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281030.png" />-cochain of a simplicial complex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281031.png" /> with coefficients in an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281032.png" /> is defined as a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281034.png" /> is the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281035.png" />-chains of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281036.png" />, i.e. the group of formal finite sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281038.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281039.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281040.png" />-simplices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281041.png" />. In particular, a cochain in the sense of Aleksandrov–Čech in an arbitrary topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281042.png" /> is a cochain of the nerve of an open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281043.png" />.
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In this situation the mapping $\delta$ on the group $C^\star$ is defined as the adjoint of $\partial$, namely the following relation
 +
\[
 +
(\delta f) (\sigma) = f (\partial \sigma)
 +
\]
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holds for every element $f\in C^\star$ and any element $\sigma\in C_\star$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281044.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281045.png" />-complex (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281046.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281047.png" />-skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281048.png" />), then the Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281049.png" /> is called the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281050.png" />-dimensional cellular cochains of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281051.png" />. The coboundary homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281052.png" /> is put equal to the connecting mappings of the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281053.png" />.
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The most common choice of coefficient group is $\mathbb Z$.  
  
In practice, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281054.png" /> is frequently provided with an additional multiplicative structure, i.e. it is a graded algebra. In these cases the coboundary mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281055.png" /> possesses the Leibniz property: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281056.png" />, where the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281057.png" /> is assumed to be homogeneous of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022810/c02281058.png" />. An example of such a graded cochain algebra is the algebra of differential forms on a smooth manifold, in which the exterior differential acts as coboundary.
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====Product structure====
 +
In practice, the group $C^\star$ is frequently provided with an additional multiplication, which makes $C^\star$ a [[Graded algebra|graded algebra]], namely the product of two homogeneous elements $\alpha$ and $\beta$ of degree $i$ and $j$ is an homogeneous element of degree $i+j$. In these cases the coboundary mapping $\delta$ possesses the Leibniz property, namely the identity
 +
\[
 +
\delta (fg) = (\delta f) g + (-1)^{{\rm deg}\, (f)} f \delta g\,  
 +
\]
 +
holds for any homoegeneous $f$ and any $g$.  
  
====References====
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===Examples===
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.J. Hilton,   S. Wylie,   "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press  (1960)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,   "Homology" , Springer  (1963)</TD></TR></table>
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Common examples of cochains are the following.
 +
* Singular cochains in a topological space $X$. Given any abelian group $G$, such cochain group is defined in duality with the group $C_\star (X, G)$ of singular chains, the Abelian group of formal finite sums $\sum_i \alpha_i \sigma_i$, where $\alpha_i\in G$ and the $\sigma_i$ are arbitrary singular simplices in $X$, i.e. continuous mappings of the [[Simplex|standard simplex]] into $X$. A singular cochain in $X$ with coefficients in $G$ is then an homogeneous element of the group ${\rm Hom}\, (C_\star (X, G), G)$.
 +
* A simplicial $n$-cochain of a simplicial complex $X$ with coefficients in an Abelian group $G$ is defined as a homomorphism $f:C_n (X) \to G$, where $C_n (X)$ is the group of $n$-chains of $X$, i.e. the group of formal finite sums $\sum_i \alpha_i \sigma_i$, where $\alpha_i\in G$ and the $\sigma_i$ are $n$-simplices in the complex $X$. In particular, a cochain in the sense of Aleksandrov–Čech in an arbitrary topological space $X$ is a cochain of the nerve of an open covering of $X$ (see [[Simplicial complex]]).
 +
* If $X$ is a [[CW-complex]] (and $X_n$ denotes the $n$-skeleton of $X$), then the Abelian group $H^n (X_n, X_{n-1})$ is called the group of $n$-dimensional cellular cochains of the complex $X$. The coboundary homomorphism $\delta: H^n (X_n, X_{n-1})\to H^{n+1} (X_{n+1}, X_n)$ is put equal to the connecting mappings of the triple $(X_{n+1}, X_n, X_{n-1})$.
 +
* If $X$ is a smooth manifold, the space of smooth [[Differential form|differential forms]] is easily seen to be a cochain complex where the coboundary is given by the usual exterior differential. The wedge product induces a product structure for which the Leibniz rule mentioned above holds.
  
 
+
===References===
 
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{|
====Comments====
+
|-
 
+
|valign="top"|{{Ref|HS}}|| P.J. Hilton,  S. Wylie,  "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press  (1960)
 
+
|-
====References====
+
|valign="top"|{{Ref|ML}}|| S. MacLane,  "Homology" , Springer  (1963)
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.E. Steenrod,  S. Eilenberg,  "Foundations of algebraic topology" , Princeton Univ. Press  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.R.F. Maunder,  "Algebraic topology" , v. Nostrand-Reinhold (1970)</TD></TR></table>
+
|-
 +
|valign="top"|{{Ref|Ma}}||  C.R.F. Maunder,  "Algebraic topology" , v. Nostrand-Reinhold (1970)
 +
|-
 +
|valign="top"|{{Ref|Sp}}|| E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)
 +
|-
 +
|valign="top"|{{Ref|SE}}|| N.E. Steenrod,  S. Eilenberg,  "Foundations of algebraic topology" , Princeton Univ. Press (1966)
 +
|-
 +
|}

Latest revision as of 16:48, 10 February 2014

2020 Mathematics Subject Classification: Primary: 18G35 Secondary: 55Nxx [MSN][ZBL]

Definition and properties

A homogeneous element of an Abelian cochain group $C^\star$ (or, in the general case, a module). A cochain group $C^\star$ is a graded Abelian group, which means that $C^\star$ is decomposed as the direct sum of subgroups $A_k$, indexed with $k\in \mathbb Z$, some of which might be trivial; an homogenenous element $f$ is an element belonging to some $A_k$, where $k$ is called the degree of the element and denoted by ${\rm deg}\, (f)$. The cochain group is also equipped with an endomorphism $\delta: C^\star \to C^\star$ of degree $+ 1$, namely mapping elements of $A_k$ into elements of $A_{k+1}$, such that $\delta \circ \delta =0$. The endomorphism $\delta$ is called the coboundary mapping or the coboundary.

Duality

A cochain group $C^\star$ arises often as dual of a chain group $C_\star$ with coefficients group $G$, i.e. as a group $C^\star = {\rm Hom} (C_\star, G)$, where

  • $G$ is an arbitrary Abelian group
  • $C_\star$, the chain group, is a graded Abelian group equipped with an endomorphism $\partial$ of degree $-1$ (the boundary mapping or boundary) with $\partial \circ \partial =0$. $C_\star$ has an operation of multiplication by elements of $G$, namely a map $C_\star \times G \to C_\star$ which is an homomorphism on each factor.

In this situation the mapping $\delta$ on the group $C^\star$ is defined as the adjoint of $\partial$, namely the following relation \[ (\delta f) (\sigma) = f (\partial \sigma) \] holds for every element $f\in C^\star$ and any element $\sigma\in C_\star$.

The most common choice of coefficient group is $\mathbb Z$.

Product structure

In practice, the group $C^\star$ is frequently provided with an additional multiplication, which makes $C^\star$ a graded algebra, namely the product of two homogeneous elements $\alpha$ and $\beta$ of degree $i$ and $j$ is an homogeneous element of degree $i+j$. In these cases the coboundary mapping $\delta$ possesses the Leibniz property, namely the identity \[ \delta (fg) = (\delta f) g + (-1)^{{\rm deg}\, (f)} f \delta g\, \] holds for any homoegeneous $f$ and any $g$.

Examples

Common examples of cochains are the following.

  • Singular cochains in a topological space $X$. Given any abelian group $G$, such cochain group is defined in duality with the group $C_\star (X, G)$ of singular chains, the Abelian group of formal finite sums $\sum_i \alpha_i \sigma_i$, where $\alpha_i\in G$ and the $\sigma_i$ are arbitrary singular simplices in $X$, i.e. continuous mappings of the standard simplex into $X$. A singular cochain in $X$ with coefficients in $G$ is then an homogeneous element of the group ${\rm Hom}\, (C_\star (X, G), G)$.
  • A simplicial $n$-cochain of a simplicial complex $X$ with coefficients in an Abelian group $G$ is defined as a homomorphism $f:C_n (X) \to G$, where $C_n (X)$ is the group of $n$-chains of $X$, i.e. the group of formal finite sums $\sum_i \alpha_i \sigma_i$, where $\alpha_i\in G$ and the $\sigma_i$ are $n$-simplices in the complex $X$. In particular, a cochain in the sense of Aleksandrov–Čech in an arbitrary topological space $X$ is a cochain of the nerve of an open covering of $X$ (see Simplicial complex).
  • If $X$ is a CW-complex (and $X_n$ denotes the $n$-skeleton of $X$), then the Abelian group $H^n (X_n, X_{n-1})$ is called the group of $n$-dimensional cellular cochains of the complex $X$. The coboundary homomorphism $\delta: H^n (X_n, X_{n-1})\to H^{n+1} (X_{n+1}, X_n)$ is put equal to the connecting mappings of the triple $(X_{n+1}, X_n, X_{n-1})$.
  • If $X$ is a smooth manifold, the space of smooth differential forms is easily seen to be a cochain complex where the coboundary is given by the usual exterior differential. The wedge product induces a product structure for which the Leibniz rule mentioned above holds.

References

[HS] P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960)
[ML] S. MacLane, "Homology" , Springer (1963)
[Ma] C.R.F. Maunder, "Algebraic topology" , v. Nostrand-Reinhold (1970)
[Sp] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[SE] N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966)
How to Cite This Entry:
Cochain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cochain&oldid=31316
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article