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 2holed 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 1holed torus to
a 2holed torus, and you get a 3holed 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.
<http://www.math.hmc.edu/funfacts>.
