**Example 1:** The boundary of a directed path is the formal difference of its endpoints.

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.