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Arrangements of Submanifolds and the Tangent Bundle Complement

Drawing parallels with hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\mathcal{A}$ of locally flat, codimension-1 submanifolds that intersect like hyperplanes. To such a collection we associate two combinatorial objects: the face category and the intersection poset. We also associate a topological space to the arrangement called the tangent bundle complement. It is the complement of union of tangent bundles of these submanifolds inside the tangent bundle of the ambient manifold. Our aim is to investigate the relationship between the combinatorics of the arrangement and the topology of the complement. In particular we show that the tangent bundle complement has the homotopy type of a finite cell complex. We generalize the classical theorem of Salvetti for hyperplane arrangements and show that this particular cell complex is completely determined by the face category.