Arrangements of Submanifolds and the Tangent Bundle Complement

Details Arrangements of Submanifolds and the Tangent Bundle Complement Arrangements of Submanifolds and the Tangent Bundle Complement

2011-10-07

arxiv.org/abs/1110.1520v2

Mathematics

Algebraic Topology

32 pages, 9 figures

Scientific paper

Drawing parallels with the theory of hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold we consider a finite collection of locally flat codimension 1 submanifolds that intersect like hyperplanes. To such a collection we associate two posets: the poset of faces and the poset of intersections. We also associate a topological space to this arrangement. The complement of union of tangent bundles of these submanifolds inside the ambient tangent bundle, which we call the tangent bundle complement. Our aim is to investigate the relationship between combinatorics of the posets and topology of the complement. Using the Nerve lemma we show that the complement has the homotopy type of a finite simplicial complex. We generalize the classical theorem of Salvetti for hyperplane arrangements and show that this particular simplicial complex, called the Salvetti complex, is completely determined by the face poset. We also characterize all the connected covers of the complement, thus generalizing the work of Delucchi and Paris.

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