Connected Sums

Figure 1

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 properties of objects which do not change under continuous deformations.

How to Cite this Page:
Su, Francis E., et al. "Connected Sums." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    topology, 2-manifolds Subjects:    topology Level:    Medium Fun Fact suggested by:   Francis Su Suggestions? Use this form. 4.66 current rating Click to rate this Fun Fact...     *   Awesome! I totally dig it!     *   Fun enough to tell a friend!     *   Mildly interesting     *   Not really noteworthy and see the most popular Facts!

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