  Given a manifold with corners $X$, we associates to it the corner structure
simplicial complex $\Sigma_X$. Its reduced K-homology is isomorphic to the
K-theory of the $C^*$-algebra $\mathcal{K}_b(X)$ of b-compact operators on $X$.
Moreover, the homology of $\Sigma_X$ is isomorphic to the conormal homology of
$X$. In this note, we constract for an arbitrary abstract finite simplicial
complex $\Sigma$ a manifold with corners $X$ such that $\Sigma_X\cong \Sigma$.
As a consequence, the homology and K-homology which occur for finite simplicial
complexes also occur as conormal homology of manifolds with corners and as
K-theory of their b-compact operators. In particular, these groups can contain
torsion.
