# Chain (algebraic topology)

In algebraic topology, a simplicial *k*-**chain**
is a formal linear combination of *k*-simplices.^{[1]}

## Integration on chains

Integration is defined on chains by taking the linear combination of integrals over the simplices in the chain with coefficients 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 is a path from point to point , where
,
and
are its constituent 1-simplices, then

**Example 2:** The boundary of the triangle is a formal sum of its edges with signs arranged to make the traversal of the boundary counterclockwise.

A chain is called a **cycle** when its boundary is zero. A chain that is the boundary of another chain is called a **boundary**. Boundaries are cycles,
so chains form a chain complex, whose homology groups (cycles modulo boundaries) are called simplicial homology groups.

**Example 3:** A 0-cycle is a linear combination of points such that the sum of all the coefficients is 0. Thus, the 0-homology group measures the number of path connected components of the space.

**Example 4:** The plane punctured at the origin has nontrivial 1-homology group since the unit circle is a cycle, but not a boundary.

In differential geometry, the duality between the boundary operator on chains and the exterior derivative is expressed by the general Stokes' theorem.

## References

- ↑ Hatcher, Allen (2002).
*Algebraic Topology*. Cambridge University Press. ISBN 0-521-79540-0.