On Fredholm boundary conditions on manifolds with corners I: Global corner's cycles obstructions

TL;DR

This paper establishes a rational isomorphism between the K-theory of b-compact operators on manifolds with corners and their conormal homology, revealing obstructions to Fredholm perturbations in higher codimensions.

Contribution

It introduces an explicit topological space linking conormal homology with K-theory, extending previous low-codimension results to all codimensions.

Findings

Proves a natural rational isomorphism between K-theory and conormal homology.

Identifies conormal homology as an obstruction to Fredholm perturbations.

Provides a new method to compute higher spectral sequence differentials.

Abstract

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with corners $X$ of any codimension, there is a natural and explicit morphism $$K_*(\mathcal{K}_b(X)) \stackrel{T}{\longrightarrow} H^{pcn}_*(X,\mathbb{Q})$$ between the $K-$theory group of the algebra $\mathcal{K}_b(X)$ of $b$-compact operators for $X$ and the periodic conormal homology group with rational coeficients, and that $T$ is a rational isomorphism. As shown by the first two authors in a previous paper this computation implies that the rational groups $H^{pcn}_{ev}(X,\mathbb{Q})$ provide an obstruction to the Fredholm perturbation property for compact connected manifold with corners. The difference with respect to the previous article of the first two authors in which they solve this problem for low codimensions is that we overcome in the present article the problem of computing the higher spectral sequence K-theory differentials associated to the canonical filtration by codimension by introducing an explicit topological space whose singular cohomology is canonically isomorphic to the conormal homology and whose K-theory is naturally isomorphic to the $K-$theory groups of the algebra $\mathcal{K}_b(X)$.