Rubber Band Around a Hole

Topology · Axiom Academy

INTRO Rubber Band Around a Hole Understanding loops and homotopy through a simple metaphor Imagine you have a rubber band lying on a flat table. If you try to shrink it down to a point, you can easily do it - just pull it inward from all sides until it's tiny, then it's gone. Click the button below to watch a rubber band shrink to a point on a flat surface. Now imagine there's a hole in the middle of the table, and the rubber band is wrapped around that hole. Can you still shrink it to a point? Try it! Click below to see what happens when you try to shrink a rubber band that's looped around a hole. You just discovered one of the most important ideas in topology! The rubber band that doesn't go around the hole can shrink to nothing, but the one that does go around the hole is stuck. In topology, we say these two loops are not homotopic . A continuous deformation from one to the other would require either cutting the rubber band or passing it through the hole - both of which break the rules! The Mathematical Idea: Homotopy What you've discovered is called homotopy - the study of when one loop can be continuously deformed into another. Two loops in a space are homotopic if you can continuously deform one into the other without breaking the loop or leaving the space. If a loop can shrink to a point, it's called null-homotopic or contractible . This simple rubber band idea is the foundation of algebraic topology - one of the most beautiful areas of mathematics!

This is the written version of the interactive lesson above. See the full Topology course.