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 cycles.
Our main theorem is that, for any manifold with corners of any codimension, there is a natural and explicit morphism between the -theory group of the algebra of -compact operators for and the periodic conormal homology group with rational coefficients, and that is a rational isomorphism.
As shown by the first two authors in a previous paper, this computation implies that the rational groups provide an obstruction to the Fredholm perturbation property for compact connected manifolds with corners.
This paper differs from that previous paper, in which the problem is solved in low codimensions, in that here we overcome the problem of computing the higher spectral sequence -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 -theory is naturally isomorphic to the -theory groups of the algebra .
"On Fredholm boundary conditions on manifolds with corners I: Global corner cycles obstructions." Ann. K-Theory 6 (4) 607 - 628, 2022. https://doi.org/10.2140/akt.2021.6.607