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 textpseudodifferential operator in the associated Formula: see textcalculus 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.
Rouse et al. (Fri,) studied this question.