Conormal homology of manifolds with corners

TL;DR

This paper explores the relationship between the conormal homology of manifolds with corners and the K-theory of associated operator algebras, showing that any finite simplicial complex's homology can be realized in this setting.

Contribution

It constructs manifolds with corners for any finite simplicial complex, demonstrating the realization of simplicial homology and K-homology in the conormal homology and K-theory of these manifolds.

Findings

Homology of the corner structure complex is isomorphic to conormal homology.

K-homology is isomorphic to the K-theory of b-compact operators.

Any finite simplicial complex's homology can be realized as conormal homology of a manifold with corners.

Abstract

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.