The end sum of surfaces

TL;DR

This paper investigates the uniqueness of the end sum operation for surfaces, establishing that the outcome is uniquely determined by the chosen ends and orientability, based on a classification of noncompact surfaces.

Contribution

It extends the understanding of end sum operations to surfaces, proving a uniqueness result that differs from higher-dimensional cases by relying on surface classification.

Findings

End sum of surfaces is uniquely determined by ends and orientability.

Adding a 1-handle at infinity is classified by ends and orientability.

Results depend on noncompact surface classification.

Abstract

End sum is a natural operation for combining two noncompact manifolds and has been used to construct various manifolds with interesting properties. The uniqueness of end sum has been well-studied in dimensions three and higher. We study end sum -- and the more general notion of adding a 1-handle at infinity -- for surfaces and prove uniqueness results. The result of adding a 1-handle at infinity to distinct ends of a surface with compact boundary is uniquely determined by the chosen ends and the orientability of the 1-handle. As a corollary, the end sum of two surfaces with compact boundary is uniquely determined by the chosen ends. Unlike uniqueness results in higher dimensions, which rely on isotopy uniqueness of rays, our results rely fundamentally on a classification of noncompact surfaces.