connected sum

Let $M$ and $N$ be two $n$-manifolds. Choose points $m\in M$ and $n\in N$, and let $U,V$ beneighborhoods of these points, respectively. Since $M$ and $N$ are manifolds, we may assumethat $U$ and $V$ are balls, with boundaries homeomorphic to $(n-1)$-spheres, since this is possiblein ${ℝ}^n$. Then let $\phi :\partial U\to \partial V$ 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 $M\mathrm{♯}N$ is $M-U$ and $N-V$ glued along the boundaries by $\phi$.

That is, $M\mathrm{♯}N$ 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 $\phi (x)=y$.