Definition:Connected Sum

Definition
The connected sum of two manifolds $A^n, B^n$ of dimension $n$ is defined as follows:

Let $\Bbb D^n$ be a closed n-disk.

Let $\alpha: \Bbb D^n \to A^n$ be a continuous (or, in the case of smooth manifolds, a smooth) injection.

Let $\beta: \Bbb D^n \to B^n$ be a similar function.

Define the set:
 * $S = \paren {A^n \setminus \map \alpha {\paren {\Bbb D^n}^\circ} } \cup \paren {B^n \setminus \map \beta {\paren {\Bbb D^n}^\circ} }$

where:
 * $\setminus$ denotes set difference
 * $\paren {\Bbb D^n}^\circ$ denotes the interior of $\Bbb D^n$.

Define an equivalence relation $\sim$ on $S$ as:
 * $x \sim y \iff \paren {\paren {x = y} \lor \paren {\map {\alpha^{-1} } x = \map {\beta^{-1} } y} }$

Since the interiors of the disks were removed from the manifolds, it necessarily follows that:
 * $\map {\alpha^{-1} } x, \map {\beta^{-1} } y \in \partial \Bbb D^n$

The connected sum $A^n \# B^n$ is defined as the quotient space of $S$ under $\sim$.