An introduction to the \mathcal F-completion, the Interface construction and the Gamma set

I'll start off with a definition that might initially sound pretty abstract and maybe confusing. The point in defining things as I do is to make our lives easier later on. The goal is to focus on a symmetrical subspace of what I'll call the \mathcal F-completion. This subspace is called \mathcal I. So let's begin: Let X be a smooth, open, geodesically convex n -dimensional manifold with regular polytope boundary, and let V = \{v_i\} be a finite set of vertices in \partial X . A block is defined as \mathcal{B} := X \cup \partial X , where \partial X denotes the boundary. A foliational completion (or \mathcal{F}-completion) of a block \mathcal{B} is a structure obtained by extending foliations to accumulate at specific vertices in the boundary. Formally, for each pair of distinct vertices (v_i, v_j) \in V \times V , there exists a foliation \mathcal{F}_{v_{ij}} , whose leaves accumulate at v_i and v_j . The $\m...