2025/01/09 by Paulo Carrillo Rouse, Rouse, Paulo Carrillo, Jean-Marie Lescure +1 · 1 voice · 1 citation
Mathematics · #19K56 #46L80 #58J32 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #math.KT #math.OA
paper · pdf · doi:10.48550/arxiv.2501.05071
openalex publication_date 2025/01/09 · arxiv published 2025/01/09 · arxiv updated 2025/01/09 · openalex created_date 2025/01/11 · openalex updated_date 2026/07/28
Given a connected manifold with corners X 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 Hcn_*(X). Using our previous works we define an index morphism K0(bT^*X)\stackrelIndev,cnX\longrightarrowHevcn(X) for X a manifold with corners of codimension less or equal to three and called here the even conormal index morphism. In the case that X is compact and connected and D is an elliptic b-pseudodifferential operator in the associated b-calculus of X we know, by our previous works and other authors works, that, up to adding an identity operator, D can be perturbed (with a regularizing operator in the calculus) to a Fredholm operator iff Indev,cnX([σD]) (where [σD]∈ K0(bT^*X) is the principal symbol class) vanishes in the even conormal homology group Hevcn(X). The main result of this paper is the explicit computation of the even and odd conormal index morphisms Indev/odd,cnX(σ)∈ Hev/oddcn(X) for X a manifold with corners of codimension less or equal to three. The coefficients of the conormal corner cycles Indev/odd,cnX(σ) are given in terms of some suspended Atiyah-Singer indices of the maximal codimension faces of X and in terms of some suspended Atiyah-Patodi-Singer indices of the non-maximal codimension faces of X. As a corollary we give a complete caracterization 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.