Circle Decompositions of Surfaces

TL;DR

This paper classifies all connected surfaces that can be decomposed into topological circles, showing that only seven specific types qualify, and establishes properties of such decompositions.

Contribution

It provides a complete classification of surfaces admitting circle decompositions and proves that all such decompositions are upper semicontinuous.

Findings

Seven surfaces admit circle decompositions.

Circle decompositions are upper semicontinuous.

Classification based on Euler characteristic and boundary components.

Abstract

We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a byproduct, we get that any circle decomposition of a surface is upper semicontinuous.