Fredholm anomalies on manifolds with corners of low codimensions and conormal corner cycles

Given a connected manifold with (embedded) corners [Formula: see text] 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, these conormal homology groups are denoted by [Formula: see text]. Using our previous works we define an index morphism [Formula: see text] for [Formula: see text] a manifold with corners of codimension less or equal to three and called here the even conormal index morphism. In the case that [Formula: see text] is compact and connected and [Formula: see text] is an elliptic [Formula: see text]pseudodifferential operator in the associated [Formula: see text]calculus of [Formula: see text] we know, by our previous works and other authors works, that, up to adding an identity operator, [Formula: see text] can be perturbed (with a regularizing operator in the calculus) to a Fredholm operator iff [Formula: see text] (where [Formula: see text] is the principal symbol class) vanishes in the even conormal homology group [Formula: see text]. The main result of this paper is the explicit computation of the even and odd conormal index morphisms [Formula: see text] for [Formula: see text] a manifold with corners of codimension less or equal to three. The coefficients of the conormal corner cycles [Formula: see text] are given in terms of some suspended Atiyah-Singer indices of the maximal codimension faces of [Formula: see text] and in terms of some suspended Atiyah-Patodi-Singer indices of the non-maximal codimension faces of [Formula: see text]. As a corollary we give a complete characterization to the obstruction of the Fredholm perturbation property for closed manifolds with corners of codimension less or equal to three in terms of the above mentioned indices of the faces, this allows us as well to give such a characterization in terms of the respective topological indices.