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 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 A_k to A_k+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

${\begin{aligned}\partial _{1}c&=\partial _{1}(t_{1}+t_{2}+t_{3})\\&=\partial _{1}(t_{1})+\partial _{1}(t_{2})+\partial _{1}(t_{3})\\&=\partial _{1}([v_{1},v_{2}])+\partial _{1}([v_{2},v_{3}])+\partial _{1}([v_{3},v_{4}])\\&=([v_{2}]-[v_{1}])+([v_{3}]-[v_{2}])+([v_{4}]-[v_{3}])\\&=[v_{4}]-[v_{1}].\end{aligned}}$

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.

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