Conormal homology of manifolds with corners

Abstract

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 $C^{\ast }$-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.