# connected sum

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 $\mathbb{R}^{n}$. Then let $\varphi:\partial U\to\partial V$ 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 $\varphi$.

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 $\varphi(x)=y$.

Title connected sum ConnectedSum1 2013-03-22 13:17:59 2013-03-22 13:17:59 bwebste (988) bwebste (988) 6 bwebste (988) Definition msc 57-00