From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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.


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 $g$.


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.


[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:
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article