Boundary operator

The operator $\partial$ that sends a chain to the chain of its oriented edges -- and satisfies $\partial(\partial c)=0$ always.

The boundary operator $\partial$ sends a $k$ -chain to the $(k{-}1)$ -chain of its oriented boundary cells. Applied twice it always gives zero, $\partial(\partial c)=0$ -- every interior edge of a subdivided region is shared by two cells with opposite orientation and cancels, leaving only the true outer boundary.

Examples

The boundary of a single oriented triangle is the signed sum of its three oriented edges.