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 $\\varphi(x)=y$ .
随便看	derivative of Riemann integral DerivativeOfXn derivative of x^n DerivativeOperatorIsUnboundedInTheSupNorm derivative operator is unbounded in the sup norm DerivativesByPureAlgebra derivatives by pure algebra DerivativesOfHyperbolicFunctions derivatives of hyperbolic functions DerivativesOfSineAndCosine derivatives of sine and cosine DerivativesOfsinXAndcosX DerivativesOfSolutionOfFirstOrderODE derivatives of solution of first order ODE derivatives of s⁢i⁢nx and c⁢o⁢sx DerivedBooleanOperations derived Boolean operations DerivedSet derived set DerivedSubgroup derived subgroup Deriving the trigonometric addition formulae using area and cosine rule DerivingTheTrigonometricAdditionFormulaeUsingAreaAndCosineRule1 DesarguesTheorem Desargues’ theorem