Fredholm Realizations of Elliptic Symbols on Manifolds with Boundary Ii: Fibered Boundary

one can quantize an invertible symbol σ ∈ C∞(S∗X; hom(π∗E, π∗F )) as an elliptic pseudodifferential operator in the Φ-b or edge calculus, ΨΦ-b(X;E,F ), introduced in [7] or alternately as an elliptic operator in the Φ-c or φ calculus, ΨΦ-c(X;E,F ), introduced in [8]. As on a closed manifold, either of these operators will induce a bounded operator acting between natural L-spaces of sections but, in contrast to closed manifolds, these operators need not be Fredholm. A well-known analysis of Atiyah and Bott established that in order to quantize σ as an operator on X for which Fredholm local boundary conditions exist, it is necessary and sufficient that the Atiyah-Bott obtruction of σ vanish, i.e., [σ] ∈ ker ( Kc(T ∗X)→ K c (T ∗∂X) ) .