Quotient Topology 3

This exercise combines thinking about quotient topologies and images/preimages. 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: after 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
• Paper towels or tissues, a couple per group
• Rubbing alcohol
• A large container such as a box or bag to carry everything to class
• This worksheet
• LaTeX file
• 2-21-1.pdf
• 2-21-2.pdf
• 2-21-3.pdf
• 2-21-4.pdf

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, paper towels, and whiteboard markers, to bring to class.
4. Print and make copies of the worksheet

In Class

1. Before class, deposit one of each item at each group area.
2. Hand out the worksheet and do your group work thing! Let students know they can clean off the inflatables between problems using the paper towels.

After Class

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