**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 of *M* # *N* is the sum of the Euler characteristics of the summands, minus two:

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.

Read more about this topic: Surface

### Famous quotes containing the words sums and/or connected:

“At Timon’s villalet us pass a day,

Where all cry out,What *sums* are thrown away!’”

—Alexander Pope (1688–1744)

“Before I had my first child, I never really looked forward in anticipation to the future. As I watched my son grow and learn, I began to imagine the world this generation of children would live in. I thought of the children they would have, and of their children. I felt *connected* to life both before my time and beyond it. Children are our link to future generations that we will never see.”

—Louise Hart (20th century)