An Introduction to Topology

Linda Green

Bay Area Circle for Teachers2012 Summer Workshop

Images from virtualmathmuseum.org

Topics

• What is topology?• Surfaces and gluing diagrams• The universe

• A math auction

Most of this material is from The Shape of Space by Jeff Weeks.

Geometry vs. Topology• Study of rigid objects• Rotations and reflections

are okay• No stretching allowed

• Study of flexible objects• Rotations and reflections are

okay• Stretching and bending are

okay• No tearing or gluing allowed

Topology vs. GeometryInformal Definitions

• The properties of an object that stay the same when you bend or stretch 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, gluing, or other violent actions.

• The properties of an object that change when you bend or stretch it 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.

Image from http://en.wikipedia.org/wiki/Topology

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?

Don’t glue the interior parts of the square together, just the edges!

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 (no winner)?

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

• Where should X go to win?

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?• Is it possible to get a Cat’s Game?

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?

Image from Plus Magazine http://plus.maths.org/content/os/issue26/features/mathart/applets2/experiments

• We have seen that the following gluing diagrams describe a topological sphere, torus, and Klein bottle.

• How many essentially different ways are their to glue the edges of a square in pairs? How many different topological surfaces result?

Surfaces made from a square

Surfaces made from a square

• Which of these surfaces are orientable?• Which represent a torus? A sphere? A Klein bottle? Something

new?

Two gluing diagrams can represent the same surface

• Try cutting apart surface E along a diagonal and regluing it to get one of the other surfaces.

• Try the same thing with surface F. • Make sure to label the edges of the cut you make to keep

track of which new edges get glued to each other.

The Projective Plane

Is there an algorithm to tell which gluing diagrams represent which surfaces?

• Look at what happens if you walk around the corners of the square.• In some gluing diagrams, all four corners glue up to form only one piece of

the surface. In other diagrams, the corners glue in pairs.

• If you mark the 4 edges and 4 vertices of the square, each pair of edges glues up so that there are only 2 distinct edges on the glued up surface. Some vertices also get glued up to each other, so there may be only 1 or 2 distinct vertices in the final glued up surface.

• Label each diagram with the number of edges and vertices AFTER gluing.

Edges, vertices, and faces

TorusV = 1E = 2F = 1

Klein bottleV = 1E = 2F = 1

Projective planeV = 2E = 2F = 1

Projective planeV = 2E = 2F = 1

SphereV = 3E = 2F = 1

Klein bottleV = 1E = 2F = 1

What surfaces do these gluing diagrams represent?

Compute the number of edges and vertices after gluing for each of these diagrams.

Euler number• All gluing diagrams that represent the same surface have the

same relationship between number of vertices, edges, and faces Gluing

diagramSurface Number

of verticesNumber of edges

Number of faces

Euler number

A

B

C

D

E

F

G

H

I

J

K

Euler number + orientability is enough

• Any two gluing diagrams that represent the same surface have the same Euler number V – E + F

• Amazingly, it is also true that any two gluing diagrams that have the same Euler number and are both orientable represent the same surface!

• Similarly, any two gluing diagrams that have the same Euler number and are both non-orientable represent the same surface!

• So Euler number plus orientability is enough to determine the surface

Gluing diagram puzzles

• Find a gluing diagram, in which all edges get glued in pairs, that does not represent a sphere, torus, Klein bottle, or projective plane

• What is the Euler number for these gluing diagrams? What surfaces do they represent?

References and Resources

• The Shape of Space by Jeff Weeks• Torus games

http://www.geometrygames.org/TorusGames/

• Gluing diagram animations http://math.arizona.edu/~rta/013/bethard.steven/construct.html

• Klein bottle experimentshttp://plus.maths.org/content/os/issue26/features/mathart/applets2/experiments

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.