On the number of connected components in complements to arrangements of submanifolds

TL;DR

This paper establishes a sharp lower bound on the number of connected components in the complement of arrangements of codimension-one submanifolds in closed manifolds, linking it to the number of submanifolds and the homology of the manifold.

Contribution

It provides a new lower bound for the number of connected components in complements of submanifold arrangements, connecting topological invariants with combinatorial properties.

Findings

Derived a sharp lower bound for the number of connected components

Analyzed possible f-values for hyperplane arrangements in projective spaces

Studied arrangements of subtori in d-dimensional tori

Abstract

We consider arrangements of n connected codimensional one submanifolds in closed d-dimensional manifold M. Let f be the number of connected components of the complement in M to the union of submanifolds. We prove the sharp lower bound for f via n and homology group H_{d-1}(M). The sets of all possible f-values for given n are studied for hyperplane arrangements in real projective spaces and for subtori arrangements in d-dimensional tori.