On the number of connected components of complements to arrangements of subtori

TL;DR

This paper investigates the possible counts of connected components in the complement of arrangements of codimension-one subtori within a d-dimensional torus, providing a comprehensive characterization for any arrangement.

Contribution

It characterizes all possible numbers of connected components in the complement for arrangements of subtori in a flat torus, regardless of arrangement specifics.

Findings

Complete set of possible component counts identified

Results apply to any arrangement of subtori

Provides a framework for understanding torus arrangements

Abstract

We consider the arrangements of subtori in a flat d - dimensional torus T. Let us consider an arrangement on n subtori of codimension one, let f be the number of connected components of the complement in T to the union of subtori. We found the set of all possible numbers f for given n and d and arbitrary arrangements of subtori.