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:
 * $S = \left({A^n \setminus \alpha \left({\left({\Bbb D^n}\right)^\circ}\right)}\right) \cup \left({B^n \setminus \beta \left({\left({\Bbb D^n}\right)^\circ}\right)}\right)$

where:
 * $\setminus$ denotes set difference
 * $\left({\Bbb D^n}\right)^\circ$ denotes the interior of $B^n$.

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

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

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