Chain (algebraic topology)
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In algebraic topology, a simplicial k-chain is a formal linear combination of k-simplices.
[edit] 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.
[edit] 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 ![t_1=[v_1, v_2]\,](http://upload.wikimedia.org/math/b/6/e/b6e7d3ecaef478651e0d87ae5d7079ad.png)
, ![t_2=[v_2, v_3]\,](http://upload.wikimedia.org/math/5/e/e/5ee705b1ad0e6ff83958b377a9ae1e89.png)
and ![t_3=[v_3, v_4]\,](http://upload.wikimedia.org/math/0/9/1/09108c28ba3db9561bd3366c94efc7e7.png)
are its constituent 1-simplices, then
![\begin{align}
\partial_1 c
&= \partial_1(t_1 + t_2 + t_3)\\
&= \partial_1(t_1) + \partial_1(t_2) + \partial_1(t_3)\\
&= \partial_1([v_1, v_2]) + \partial_1([v_2, v_3]) + \partial_1([v_3, v_4]) = \\
&= ([v_2]-[v_1]) + ([v_3]-[v_2]) + ([v_4]-[v_3]) = \\
&= [v_4]-[v_1].
\end{align}](http://upload.wikimedia.org/math/5/7/8/5781838f721c900ee97394ad2ac1b1a8.png)
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.
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