Conormal homology of manifolds with corners

Given a manifold with corners $X$, we associate 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 construct 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.