By [NX13, Theorem 4.2.4], the ${\mathbb{Z}}$-PA-space $\operatorname{Sk}(X)$ is connected, of pure dimension $d$, and is a deformation retract of $X^{\mathrm{an}}$. Further, $\operatorname{Sk}(X)$ is a pseudomanifold with boundary, i.e. for some (or, equivalently, any) triangulation $\Delta$ of $\operatorname{Sk}(X)$, we have:

• (a)

  Non-branching property: every $(d-1)$-simplex of $\Delta$ is contained in at most two $d$-simplices

• (b)

  Strong connectedness: every pair of $n$-simplices $\sigma$, $\sigma^{\prime}$ is joined by a chain of $n$-simplices $\sigma=\sigma_{1},\dots,\sigma_{N}=\sigma^{\prime}$ with $\sigma_{i}$ and $\sigma_{i+1}$ sharing a common $(n-1)$-face.

In the maximally degenerate case $d=n$, $\operatorname{Sk}(X)$ is even a pseudomanifold, i.e. (a) is replaced by

• (a’)

  every $(n-1)$-simplex of $\Delta$ is contained in exactly two $n$-simplices.

See also [KX15] for even more precise results on the structure of $\operatorname{Sk}(X)$. For example, $\operatorname{Sk}(X)$ is homeomorphic to a sphere if $n\leq 3$.