Let $X$ be a geometrically connected, smooth and proper $K$-variety with trivial canonical sheaf. Then the following properties hold.

1. (1)

   The essential skeleton $\mathrm{Sk}(X)$ is a strong deformation retract of $X^{\mathrm{an}}$.

2. (2)

   If $\mathscr{X}$ is a proper $snc$-model of $X$ over $R$, then $\mathrm{Sk}(X)$ is contained in $\mathrm{Sk}(\mathscr{X})$ and can be obtained from $\mathrm{Sk}(\mathscr{X})$ (as a topological subspace of $\mathrm{Sk}(\mathscr{X})$ with piecewise affine structure) by a finite number of elementary collapses.

3. (3)

   The essential skeleton $\mathrm{Sk}(X)$ is a pseudo-manifold with boundary. If $k$ is algebraically closed and $\mathrm{Sk}(X)$ has dimension $\mathrm{dim}(X)$, then it is a closed pseudo-manifold.

4. (4)

   Assume that $k$ is algebraically closed and $X$ is projective. Let $\sigma$ be a topological generator of the absolute Galois group $G(K^{a}/K)$ and let $\ell$ be a prime. Then $\mathrm{Sk}(X_{K})$ has dimension $n=\dim(X)$ if and only if the action of $\sigma$ on

   $$H^{n}_{\mathrm{\acute{e}t}}(X\times_{K}K^{a},\mathbb{Q}_{\ell})$$

   has a Jordan block of size $n+1$. If this holds, and $h^{i,0}(X)=0$ for $0<i<n$, then $\mathrm{Sk}(X_{K})$ is a $\mathbb{Q}$-homology sphere.