Arrangements of Submanifolds and the Tangent Bundle Complement

Abstract

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.

30 pages, current version is a major revision; the main result is extended to a more general setting using the language of cellular stratified spaces and acyclic categories. Typos fixed and made some stylistic changes

References in corpus (4)

• Braid groups and Artin groups
• Cellular Stratified Spaces II: Basic Constructions
• Cellular Stratified Spaces I: Face Categories and Classifying Spaces
• The integer cohomology of toric Weyl arrangements

Cited by in corpus (5)

• Faithful Actions from Hyperplane Arrangements
• On a Generalization of Zaslavsky's Theorem for Hyperplane Arrangements
• On the number of connected components in complements to arrangements of submanifolds
• On the number of connected components of complements to arrangements of subtori
• On Arrangements of Pseudohyperplanes