## Connected sums::Surface

### ::concepts

Surface::surfaces    Which::closed    Sphere::space    Boundary::plane    Torus::compact    '''P'''::complex

Connected sums The connected sum of two surfaces M and N, denoted M # N, is obtained by removing a disk from each of them and gluing them along the boundary components that result. The boundary of a disk is a circle, so these boundary components are circles. The Euler characteristic $\chi$ of M # N is the sum of the Euler characteristics of the summands, minus two:

$\chi(M \# N) = \chi(M) + \chi(N) - 2.\,$

The sphere S is an identity element for the connected sum, meaning that S # M = M. This is because deleting a disk from the sphere leaves a disk, which simply replaces the disk deleted from M upon gluing.

Connected summation with the torus T is also described as attaching a "handle" to the other summand M. If M is orientable, then so is T # M. The connected sum is associative, so the connected sum of a finite collection of surfaces is well-defined.

The connected sum of two real projective planes, P # P, is the Klein bottle K. The connected sum of the real projective plane and the Klein bottle is homeomorphic to the connected sum of the real projective plane with the torus; in a formula, P # K = P # T. Thus, the connected sum of three real projective planes is homeomorphic to the connected sum of the real projective plane with the torus. Any connected sum involving a real projective plane is nonorientable.

Surface sections
Intro  Definitions and first examples  Extrinsically defined surfaces and embeddings  Construction from polygons  Connected sums   Closed surfaces   Surfaces in geometry  See also  Notes  References  External links

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