Fibered Cusp b-Pseudodifferential Operators and its Applications

Abstract

Let $X$ be a smooth compact manifold with corners which has two embedded boundary hypersurfaces ${\partial }_{0}X,{\partial }_{1}X$, and a fiber bundle $\varphi :{\partial }_{0}X\to Y$ is given. By using the method of blowing up, we define a pseudodifferential culculus ${\Psi }_{\Phi ,b}^{\ast }(X)$ generalizing the $\Phi$-calculus of Mazzeo and Melrose and the (small) $b$-calculus of Melrose. We discuss the Fredholm condition of such operators and prove the relative index theorem. And as its application, the index theorem of "non-closed" $\mathbb{Z}/k$ - manifolds is proved.