A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle, a torus, a 2-holed torus are all examples of surfaces. We do not distinguish between a sphere and a deformed sphere… we say they are “topologically equivalent”.

You know how to add numbers. But did you know that there is a way to add surfaces? It’s called the “connect sum”. To connect sum two surfaces you pull out a disc from each, creating “holes”, and then sew the two surfaces together along the boundaries of the holes. This gives another surface! Connect sum a 1-holed torus to a 2-holed torus, and you get a 3-holed torus. Connect sum a projective plane with a projective plane, and you get a Klein+bottle! And, it can be shown that if you connect sum three projective planes it is the same surface as the connect sum of a torus and one projective plane!

The operation is commutative, associative and there is even an identity element: just add a sphere to any surface and you get back that surface!

But there is no “inverse” operation: you cannot connect sum a torus to anything and hope to get a sphere…

**Presentation Suggestions:**

Draw some fun pictures to illustrate.

**The Math Behind the Fact:**

This belongs to a field of mathematics known as topology, which, loosely speaking, is the study of continuous functions and properties of objects which do not change under continuous deformations.

**How to Cite this Page:**

Su, Francis E., et al. “Connected Sums.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**Fun Fact suggested by:**

Francis Su