Chain (algebraic topology)

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This article is about algebraic topology. For the term chain in order theory, see chain (order theory).

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

Integration on chains[edit]

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[edit]

The boundary of a polygonal curve is a linear combination of its nodes; in this case, some linear combination of A1 through A6. Assuming the segments all are oriented left-to-right (in increasing order from Ak to Ak+1), the boundary is A6 − A1.
A closed polygonal curve, assuming consistent orientation, has null boundary.

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\, is a path from point v_1\, to point v_4\,, where t_1=[v_1, v_2]\,, t_2=[v_2, v_3]\, and t_3=[v_3, v_4]\, are its constituent 1-simplices, then

\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].

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.