# An Introduction to Topology Linda Green - Marin Math Circle by hcj

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```									An Introduction to Topology
Linda Green
Nueva Math Circle
September 30, 2011

Images from virtualmathmuseum.org
Topics
•   The universe
•   Definitions
•   Surfaces and gluing diagrams
•   The universe
The Universe
• Is the universe finite or infinite?
• If we could step outside of it, what would it
look like? What is its shape?

Many of the ideas in this talk are explored in more detail in The Shape of Space by Jeff Weeks
Dimension
Informal Definitions
•  1-dimensional: Only one number is required to specify a
location; has length but no area. Each small piece looks like a
piece of a line.
• 2-dimensional: Two numbers are required to specify a
location; has area but no volume. Each small piece looks like a
piece of a plane.
• 3-dimensional: Three numbers are required to specify a
location; has volume. Each small piece looks like a piece of
ordinary space.
Topology vs. Geometry
• The properties of an object that stay the same
when you bend, stretch or twist it are called the
topology of the object. Two objects are
considered the same topologically if you can
deform one into the other without tearing,
cutting, pinching, gluing, or other violent actions.
• The properties of an object that change when
you bend, stretch, or twist are the geometry of
the object. For example, distances, angles, and
curvature are parts of geometry but not topology.
Deforming an object doesn’t change
it’s topology

A topologist is someone who can’t tell the difference between a coffee cup and a doughnut.
Which surfaces are topologically the
same?
Gluing diagrams
• What topological surface do you get when you
glue (or tape) the edges of the triangle
together as shown?
Gluing diagrams
• What do you get when you glue the edges of
the square together like this?
Gluing diagrams
• What surface is this?

S2 (a sphere)

• And this?

T2 (a torus)
Life inside the surface of a torus
• What happens as this 2-dimensional creature
travels through its tiny universe?

What does it see when it looks forward? Backward?
Left? Right?
Tic-Tac-Toe on a Torus
• Where should X go to win?

What if it is 0’s turn?
Tic-Tac-Toe on the Torus
• Which of the following positions are equivalent in torus tic-tac-
toe?

• How many essentially different first moves are there in torus
tic-tac-toe?
• Is there a winning strategy for the first player?
• Is it possible to get a Cat’s Game?
Another surface
• What surface do you get when you glue
together the sides of the square as shown?

K2 (a Klein bottle)
Life in a Klein bottle surface
What happens as this creature travels through its
Klein bottle universe?

• A path that brings a traveler back to his starting point
mirror-reversed is called an orientation-reversing path.
How many orientation-reversing paths can you find?
• A surface that contains an orientation-reversing path is
called non-orientable.
Tic-Tac-Toe on a Klein bottle
• How many essentially different first moves are
there in Klein bottle tic-tac-toe?

• Is there a winning strategy for the first player?
What happens when you cut a Klein
bottle in half?
• It depends on how you cut it.

Cutting a Klein bottle

Another Klein bottle video
Three dimensional spaces
• How can you make a 3-dimensional universe
that is analogous to the 2-dimensional torus?

• Is there a 3-dimensional analog to the Klein
bottle?
Name that Surface
• What two surfaces do these two gluing
diagrams represent?
What topological surface is this?

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