Quotient Topology

When I first studied topology, I had a hard time merging the formal notion of quotients to the idea of "gluing" pieces together. I also really wanted to bring toys to class. The inflatable props used in this activity can be reused to discuss fundamental groups.

Timing

• Prep time: 30 minutes
• Class time: 20 minutes (variable)
• Curriculum location: introducing the quotient topology

Supplies

• Colored paper
• Scotch tape
• Small inflatable balls such as these, one per group
• Small inflatable rings such as these (minus the antlers), one per group
• A paper cutter/scissors
• Hand air pump (or strong lungs)
• White board markers, one per group
• Rubbing alcohol
• A large container such as a box or bag to carry everything to class

Note: You can always buy demonstration-sized tori and spheres as well, but having these will significantly increase your prep time because they take much longer to inflate.

Instructor Instructions

Preparation

1. Cut paper into two inch strips and form Mobius bands (you may want to attach two together to make it longer)
2. Inflate balls and rings
3. Pack Mobius bands, spheres, tori, and whiteboard markers to bring to class.

In Class

1. Before class, deposit one of each item at each group area.
2. If you haven't already, introduce the formal and intuitive ideas of the quotient topology.
3. Draw a unit square on the board, with vertices at (0,0), (0,1), (1,0), and (1,1). Tell the students that each of the shapes before them - a sphere, a torus, and a mobius strip - can be realized as a quotient of the square. Their job is the figure out the equivalence classes in the quotient space that makes it possible to do so.
4. Do your group work thing! I've found it helpful to encourage the students to bend and fold their notebook paper, and draw on the inflatables with the whiteboard markers, to help them visualize as I walk around.
5. After sufficient time has passed, have the students write their answers on the board (or call on different groups who had different correct answers) so the class can compare.

After Class

• Clean the whiteboard marker off the inflatables immediately. If the ink sticks, use the rubbing alcohol to get the remainder off.