See that \(z=\partial \Delta^{\, n+1}\), where \(\partial \Delta^{\, n+1}\) is called the \(n\)-dimensional tetrahedron \(\tau^{\, n}\), is the unique \((n-1)\)-cycle up to a scalar on \(\partial \Delta^{\, n}\) as a simplicial complex. Since \(C^{\, n+1}\, (\tau^{\, n})=0\), it also generates the simplicial cohomology group \(C^{\, n}(\tau^{\, n})=H^{\, n}(\tau^{\, n})\simeq \mathbb{Z}\).