Cycle (algebraic topology)

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

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.

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 ${\displaystyle c=t_{1}+t_{2}+t_{3}\,}$ is a path from point ${\displaystyle v_{1}\,}$ to point ${\displaystyle v_{4}\,}$, where ${\displaystyle t_{1}=[v_{1},v_{2}]\,}$, ${\displaystyle t_{2}=[v_{2},v_{3}]\,}$ and ${\displaystyle t_{3}=[v_{3},v_{4}]\,}$ are its constituent 1-simplices, then

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