Connected sums of compact connected surfaces, with disk and gluing choices retained

Let $S$ and $T$ be nonempty compact connected boundaryless topological 2-manifolds (Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces). Choose embedded closed disks $D_S\subset S$ and $D_T\subset T$ whose boundaries lie in the respective coordinate charts, and choose a homeomorphism $\varphi :\mathrm{\partial }{D}_S\to \mathrm{\partial }{D}_T$. Define the connected-sum model $S{\mathrm{\#}}_{\varphi }T$ to be

$((S\setminus \mathrm{int}{D}_S)\bigsqcup (T\setminus \mathrm{int}{D}_T))\mathrm{/}\text{\hspace{0.17em}⁣}{\sim }_{\varphi }\mathrm{.}$

with the quotient topology (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection), where $x{\sim }_{\varphi }\varphi (x)$ for every $x\in \mathrm{\partial }{D}_S$ and all other points are identified only with themselves. When $S$ and $T$ are oriented, we require $\varphi$ to reverse the induced orientations of the two boundary circles. For unoriented factors, the homeomorphism $\varphi$ is part of the model data.

An iterated connected sum retains the disks, boundary maps, and parentheses used at every binary gluing. In this library, a displayed standard normal form uses the fixed polygonal model obtained by concatenating its indicated handle blocks $ab{a}^{-1}{b}^{-1}$ or crosscap blocks $aa$. No assertion that changing the disk or gluing choices preserves the homeomorphism type is included in this definition; later proofs use the explicit models they construct.