In algebraic topology, a k-chain
is a formal linear combination of the k-cells in a cell complex. In simplicial complexes (respectively, cubical complexes), k-chains are combinations of k-simplices (respectively, k-cubes), but not necessarily connected. Chains are used in homology; the elements of a homology group are equivalence classes of chains.
Contents
Definition
For a simplicial complex
X
{\displaystyle X}
, the group
C
n
(
X
)
{\displaystyle C_{n}(X)}
of
n
{\displaystyle n}
-chains of
X
{\displaystyle X}
is given by:
C
n
(
X
)
=
{
∑
i
m
i
σ
i
|
m
i
∈
Z
}
{\displaystyle C_{n}(X)=\left\{\sum \limits _{i}m_{i}\sigma _{i}|m_{i}\in \mathbb {Z} \right\}}
where
σ
i
{\displaystyle \sigma _{i}}
are singular
n
{\displaystyle n}
-simplices of
X
{\displaystyle X}
. Note that an element in
C
n
(
X
)
{\displaystyle C_{n}(X)}
is not necessarily a connected simplicial complex.
Integration on chains
Integration is defined on chains by taking the linear combination of integrals over the simplices in the chain with coefficients (which are typically integers).
The set of all k-chains forms a group and the sequence of these groups is called a chain complex.
Boundary operator on chains
The boundary of a chain is the linear combination of boundaries of the simplices in the chain. The boundary of a k-chain is a (k−1)-chain. Note that the boundary of a simplex is not a simplex, but a chain with coefficients 1 or −1 – thus chains are the closure of simplices under the boundary operator.
Example 1: The boundary of a path is the formal difference of its endpoints: it is a telescoping sum. To illustrate, if the 1-chain
c
=
t
1
+
t
2
+
t
3
{\displaystyle c=t_{1}+t_{2}+t_{3}\,}
is a path from point
v
1
{\displaystyle v_{1}\,}
to point
v
4
{\displaystyle v_{4}\,}
, where
t
1
=
[
v
1