Chain (algebraic Topology) - Boundary Operator On Chains

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, 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: 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.