Fredholm anomalies on manifold with corners of low codimensions and conormal corner cycles
Paulo Carrillo Rouse, Jean-Marie Lescure
TL;DR
The work addresses obstructions to turning elliptic $b$-pseudodifferential operators on manifolds with corners into Fredholm problems by introducing conormal homology $H^{cn}_*(X)$ and the even conormal index morphism $Ind^{X}_{cn,ev}$. It derives explicit formulas for the even/odd conormal index morphisms $Ind^{X}_{cn,ev/odd}$ in codimensions up to three, expressing coefficients through suspended Atiyah-Singer indices on maximal faces and suspended Atiyah-Patodi-Singer indices on non-maximal faces, using a framework built from the $b$-groupoid and its noncommutative tangent space. These computations yield concrete obstruction criteria for the Fredholm perturbation property, both in analytic and topological terms, and align analytic obstructions with topological indices such as $Ind^{top}_{topFred}$ and $Ind^{top}_{AS}$ through a coherent suspension formalism. The results provide a complete characterization of when (stable) Fredholm perturbations exist for low-codimension closed manifolds with corners, and establish topological reformulations of corner anomalies in codimensions 2 and 3.
Abstract
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 $H^{cn}_*(X)$. Using our previous works we define an index morphism $$K^0(^bT^*X)\stackrel{Ind_{ev,cn}^X}{\longrightarrow}H_{ev}^{cn}(X)$$ 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 $Ind_{ev,cn}^X([σ_D])$ (where $[σ_D]\in K^0(^bT^*X)$ is the principal symbol class) vanishes in the even conormal homology group $H_{ev}^{cn}(X)$. The main result of this paper is the explicit computation of the even and odd conormal index morphisms $Ind_{ev/odd,cn}^X(σ)\in H_{ev/odd}^{cn}(X)$ for $X$ a manifold with corners of codimension less or equal to three. The coefficients of the conormal corner cycles $Ind_{ev/odd,cn}^X(σ)$ 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.