Geometric obstructions for Fredholm boundary conditions for manifolds with corners

Abstract

For every connected manifold with corners we use a homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space), $χ_{cn}:=χ_0-χ_1$, is given by the alternated sum of the number of (open) faces of a given codimension. The main result of the present paper is that for a compact connected manifold with corners $X$ given as a finite product of manifolds with corners of codimension less or equal to three we have that 1) If $X$ satisfies the Fredholm Perturbation property (every elliptic pseudodifferential b-operator on $X$ can be perturbed by a b-regularizing operator so it becomes Fredholm) then the even Euler corner character of $X$ vanishes, i.e. $χ_0(X)=0$. 2) If the even Periodic conormal homology group vanishes, i.e. $H_0^{pcn}(X)=0$, then $X$ satisfies the stably homotopic Fredholm Perturbation property (i.e. every elliptic pseudodifferential b-operator on $X$ satisfies the same named property up to stable homotopy among elliptic operators). 3) If $H_0^{pcn}(X)$ is torsion free and if the even Euler corner character of $X$ vanishes, i.e. $χ_0(X)=0$ then $X$ satisfies the stably homotopic Fredholm Perturbation property. For example for every finite product of manifolds with corners of codimension at most two the conormal homology groups are torsion free. The main theorem behind the above result is the explicit computation in terms of conormal homology of the $K-$theory groups of the algebra $\mathcal{K}_b(X)$ of $b$-compact operators for $X$ as above. Our computation unifies the only general cases covered before, for codimension zero (smooth manifolds) and for codimension 1 (smooth manifolds with boundary).