## 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 <math>\chi</math> of *M* # *N* is the sum of the Euler characteristics of the summands, minus two:

- <math>\chi(M \# N) = \chi(M) + \chi(N) - 2.\,</math>

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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