  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.
