# Connected Sums

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 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 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 and properties of objects which do not change under continuous deformations.

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

Fun Fact suggested by:
Francis Su

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