# Chain (algebraic topology)

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),^{[1]}^{[2]}^{[3]} but not necessarily connected. Chains are used in homology; the elements of a homology group are equivalence classes of chains.

## Definition

[edit]For a simplicial complex , the group of -chains of is given by:

where are singular -simplices of . that any element in not necessary to be a connected simplicial complex.

## Integration on chains

[edit]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

[edit]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:** 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

[edit]**^**Hatcher, Allen (2002).*Algebraic Topology*. Cambridge University Press. ISBN 0-521-79540-0.**^**Lee, John M. (2011).*Introduction to topological manifolds*(2nd ed.). New York: Springer. ISBN 978-1441979391. OCLC 697506452.**^**Kaczynski, Tomasz; Mischaikow, Konstantin; Mrozek, Marian (2004).*Computational homology*. Applied Mathematical Sciences. Vol. 157. New York: Springer-Verlag. doi:10.1007/b97315. ISBN 0-387-40853-3. MR 2028588.

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