Definition:Connected Sum

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

Let $$\mathbb{D}^n \ $$ be an closed n-disk and let $$\alpha:\mathbb{D}^n \to A^n \ $$ be a continuous (or, in the case of smooth manifolds, a smooth) injection, and $$\beta:\mathbb{D}^n \to B^n \ $$ be a similar function.

Define the set $$S = \left({A^n - \alpha(\text{int}(\mathbb{D}^n))}\right) \cup \left({B^n - \beta(\text{int}(\mathbb{D}^n))}\right) \ $$ and define a equivalence relation $$\sim \ $$ on $$S \ $$ as

$$x \sim y \iff \left({\left({x=y}\right)\or\left({\alpha^{-1}(x) = \beta^{-1}(y)}\right)}\right) \ $$

Since the interiors of the disks were removed from the manifolds, $$\alpha^{-1}(x), \beta^{-1}(y) \in \partial \mathbb{D}^n \ $$ necessarily.

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