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 $\\mathbb{R}^{n}$ . 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 $\\varphi(x)=y$ .

单词	ConnectedSum1
释义	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 $\\mathbb{R}^{n}$ . 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 $\\varphi(x)=y$ .
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