Cochain

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)