  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 $\mathcal{K}_b(X)$.
