A text I am reading treats $PL$-manifolds as if they were pure simplicial complexes. However it defines $n$-dimensional $PL$-manifold as $n$-dimensional simplicial complex with the property that link of every vertex is a piecewise linear sphere $S^{n-1}$. How does this definition guarantee that the simplicial complex must be pure?
Thanks.
I presume that by "pure" in this context you mean that every simplex is a face of some $n$-simplex, in which case this follows from an algebraic topology calculation.
First, the condition on links implies that $X$ is an $n$-manifold in the topological sense.
In a simplicial complex $X$, if $\sigma \subset X$ is a simplex of dimension $m$, and if $\sigma$ is not a face of any higher dimensional simplex of $X$, then for any $x \in \text{interior}(\sigma)$ and any integer $i \ge 0$, the relative homology of $X$ relative to its subspace $X-x$ is given by: $$H_i(X,X-x) \approx \begin{cases} 0 & \quad\text{if $i \ne m$} \\ \mathbb Z &\quad\text{if $i=m$} \end{cases} $$ But if $X$ is an $n$-manifold, then $H_n(X,X-x) \approx \mathbb Z$.