connected sum

Let M and N be two n-manifoldsMathworldPlanetmath. Choose points mM and nN, and let U,V be neighborhoodsMathworldPlanetmathPlanetmath 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 n. Then let φ:UV 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 sumMathworldPlanetmathPlanetmath MN is M-U and N-V glued along the boundaries by φ.

That is, MN is the disjoint unionMathworldPlanetmath of M-U and N-V modulo the equivalence relationMathworldPlanetmath xy if xU, yV and φ(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