# 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 $\phi :\partial U\to \partial V$ 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 ^{}* $M\mathrm{\u266f}N$ is $M-U$ and $N-V$ glued along the boundaries by $\phi $.

That is, $M\mathrm{\u266f}N$ 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 $\phi (x)=y$.

Title | connected sum |
---|---|

Canonical name | ConnectedSum1 |

Date of creation | 2013-03-22 13:17:59 |

Last modified on | 2013-03-22 13:17:59 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 6 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 57-00 |