Boundary subcomplex of a simplex ($\partial {\Delta }^n$)

The boundary subcomplex $\partial {\Delta }^n$ consists of all proper faces of the $n$-simplex. Its geometric realization of a simplicial complex is homeomorphic to the sphere $S^{n-1}$, while $|{\Delta }^n|$ is homeomorphic to the closed disk $D^n$.