Let $M$ and $N$ be two $n$ -manifolds. Choose points $m\in M$ and $n\in N$ , and let $U, V$ be neighborhoods of these points, respectively. Since $M$ and $N$ are manifolds, we may assume that $U$ and $V$ are balls, with boundaries homeomorphic to $(n-1)$ -spheres, since this is possible in $\R^n$ . Then let $\vp:\partial U\to\partial V$ be a homeomorphism. If $M$ and $N$ are oriented, this should be orientation preserving with respect to the induced orientation (that is, degree 1). Then the connected sum $M\sharp N$ is $M-U$ and $N-V$ glued along the boundaries by $\vp$ .

That is, $M\sharp N$ is the disjoint union of $M-U$ and $N-V$ modulo the equivalence relation $x\sim y$ if $x\in\partial U$ , $y\in\partial V$ and $\vp(x)=y$ .