An Introduction to Topology

Linda Green

Nueva Math CircleSeptember 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

DimensionInformal 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?

• And this?S2 (a sphere)

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

• 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.

What happens as this creature travels through its Klein bottle universe?

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?