= Solution
Let $K_n$ be the abstract <simplex> on the vertex set
$$
\{v_0,v_1,\ldots,v_n\};
$$
thus every subset of these vertices is a simplex. Let $L_{n-1}$ be its <boundary subcomplex of a simplex>, consisting of all proper faces, so the only simplex of $K_n$ absent from $L_{n-1}$ is the full $n$-simplex $[v_0\cdots v_n]$.
The <geometric realization of a simplicial complex> $|K_n|$ is the usual solid geometric $n$-simplex, which is homeomorphic to the <closed disk> $D^n$. Under such a homeomorphism its topological boundary is $|L_{n-1}|$, and
$$
\boxed{|L_{n-1}|\cong S^{n-1}.}
$$
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